This book emphasizes some basic results in functional and classical analysis, including Hahn-Banach-type theorems, the M

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*English*
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*Year 2022*

*Table of contents : ContentsAcknowledgementsIntroductionChapter OneChapter TwoChapter ThreeChapter FourChapter FiveChapter SixChapter SevenChapter EightReferences*

Convexity, Extension of Linear Operators, Approximation and Applications

Convexity, Extension of Linear Operators, Approximation and Applications By

Octav Olteanu

Convexity, Extension of Linear Operators, Approximation and Applications By Octav Olteanu This book first published 2022 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2022 by Octav Olteanu All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-8504-2 ISBN (13): 978-1-5275-8504-1

To My Family

CONTENTS

Acknowledgements .................................................................................. viii Introduction ................................................................................................. 1 Chapter One ................................................................................................. 4 On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem Chapter Two .............................................................................................. 78 Elements of Representation Theory Chapter Three ............................................................................................ 82 Earlier and Recent Results on Convexity and Optimization Chapter Four ............................................................................................ 102 Invariant Subspaces and Invariant Balls of Bounded Linear Operators Chapter Five ............................................................................................ 114 From Linear to Sublinear and to Convex Operators Chapter Six .............................................................................................. 125 Moment Problem, Finite-simplicial Sets and Related Sandwich Results Chapter Seven.......................................................................................... 142 On Newton’s Method for Convex Functions and Operators and a Connection to Contraction Principle Chapter Eight ........................................................................................... 154 On a Class of Special Functional Equations References ............................................................................................... 165

ACKNOWLEDGEMENTS

The author would like to thank the Editor, the Designer and the Typesetting Team for excellent collaboration.

INTRODUCTION OCTAV OLTEANU

The aim of the present book is to emphasize recent results in the following actual research fields, presented in the eight chapters of the book. The main topics are: 1) Hahn-Banach type theorems and some of their motivations; the moment problem and related problems; giving a direct sharp proof for a main generalization of Hahn-Banach theorem; proving and applying polynomial approximation on unbounded subsets, that leads to existence and uniqueness of the solution for some Markov moment problems; characterizing the existence and uniqueness of the linear solution ܶ, such that ܶଵ ܶ ܶଶ on the positive cone of the domain space. Here ܶଵ and ܶଶ are given bounded linear operators; reviewing a construction of a polynomial solution for multidimensional moment problem; applying the Mazur-Orlicz theorem to concrete spaces and operators. 2) Elements of representation theory. The notion of a barycenter for a probability measure. 3) Pointing out properties, evaluating and optimizing convex functions and convex operators. 4) Constructing invariant subspaces for a large class of bounded linear operators and related results. 5) Proving results from linear analysis for convex operators; emphasizing relationship between linear and sublinear continuous operators; extending inequalities via Krein-Milman theorem. 6) Proving topological versions of sandwich theorems of type ݂ ݄ ݃, where ݂, െ݃ are convex and ݄ is affine, on bounded and unbounded special convex subsets, called finite-simplicial sets. Generally, a finitesimplicial subset of a real vector space can be unbounded in any locally convex topology on the entire space. 7) Pointing out a global Newton like method for convex functions and operators and its connection to contraction principle. 8) Proving results on special functional equations over the real and over the complex fields. All our theorems are accompanied by (or represent) examples, solving concrete problems related to basic spaces of functions and respectively equations. The interested reader can find detailed proofs of recent results, as well as of our earlier basic results. A common point of most of these themes is the notion of convex function (or operator). In this respect, the linear solution of a Markov moment problem is dominated by a convex operator and is minorated by the null operator on

2

Introduction

the convex cone of the domain space. This last property is called positivity of the solution. The dominating convex continuous operator controls the norm of the linear solution. Recall that a Markov moment problem is an interpolation problem with two constraints on the linear solution. Here the dominated operator is not necessarily null. For details, see the introductory part of the first chapter. The existence of such a solution is proved by means of Hahn-Banach type theorems and their generalizations. All necessary such earlier results are recalled without their proofs. A direct sharp proof of a basic such result is given in Chapter 1. Another important aspect of the Markov moment problem is the uniqueness of the solution. Sometimes the uniqueness follows from the proof of the existence for the solution, via polynomial approximation results. Since the values of the linear continuous solution on polynomials are prescribed, the uniqueness of the solution follows whenever the polynomials are dense in the domain function space. The most interesting case is that of function spaces in several real variables, for which polynomial approximation on arbitrary (unbounded) closed subsets holds in some ܮଵ spaces. The most interesting results hold for Cartesian product of closed intervals. In particular, it works for spaces such as ܮଵఓ (Թ ), ܮଵఓ ([0, λ) ), etc., where ߤ is a product measure: ߤ = ߤଵ × ߤ × ڮ , ߤ being a positive regular Borel ܯെdeterminate (moment determinate) measure on Թ, with finite moments Թ ݐ ݀ߤ of all orders ݇ א Գ, ݆ = 1, … , ݊, respectively ߤ is an ܯെ determinate measure on [0, λ), ݆ = 1, … , ݊. Recall that a measure ߥ on Թ is called ܯെ determinate if it is uniquely determinate by its classical moments Թ ݐ ݀ߥ, ݇ אԳ, or, equivalently, by its values on polynomials. Our polynomial approximation results partially solve the difficulty arising from the fact that there exist nonnegative polynomials on Թ , ݊ 2, which are not expressible as sums of squares. This way, some of the conditions in our statements are formulated in terms of quadratic forms. On the other hand, these approximation methods lead to a characterization of positivity of some bounded linear operators only in terms of quadratic forms. Notably, in Chapter 1 a special attention is focused on the Mazur-Orlicz theorem in concrete spaces. Many of the references of this book refer to applications related to convexity, geometric aspects in analysis and functional analysis and/or Hahn-Banach theorem. Various aspects of the moment problem are discussed in [1], [2], [4-7], [12], [13], [19], [23-25], [27-29], [31], [32], [35], [41], [45], [47-49], [51-53], [55], [58-63], [67], [70], [71], [75], [76], [78]. For some recent results and applications related to the present work, see [11], [20], [21], [27], [33], [39], [40], [59-64], [71], [74], [76]. As we will see in the introduction of the first chapter, solving many moment problems

Convexity, Extension of Linear Operators, Approximation and Applications

3

requires both Hahn-Banach type theorems and polynomial approximation on Cartesian products of closed (unbounded) intervals. This is the reason for placing the next chapter as the first one. Chapter 2 deals with elements of representation theory, while chapter 3 is devoted to evaluating and optimizing convex functions and operators, under certain convex or even linear constraints. For Pareto optimization and related applications see [9], [10]. For further results on convex functions and operators see [30], [36], [39], [43]. Chapter 4 is devoted to construction of invariant (closed) subspaces for a class of bounded linear operators. A related differential equation is studied and discussed in this respect. Invariance of the unit ball of some ܮଵ spaces is also under attention. Here polynomial approximation is applied again. It allows passing from inequalities verified on special nonnegative polynomials, to the same inequalities on arbitrary nonnegative functions in the domain space. In Chapter 5, a uniform boundedness property for classes of convex operators is proved and its relationship with previous related results in the literature is briefly discussed. As an important particular case, we point out applications of this result to classes of sublinear operators. Extending inequalities via Krein-Milman theorem is considered in the end of this chapter. The first aim of Chapter 6 is to prove topological versions for sandwich results ݂ ݄ ݃, where ݂, െ݃ are convex and ݄ is affine on ܭ. The case when ܭis a Choquet simplex, as well as that of ܭ being a finite- simplicial set [2] is under attention. Second, applications of Krein-Milman and Carathéodory’s theorem are emphasized. Chapter 7 deals with a global Newton’s method for convex monotone operators. The Newton iterations are well-known. The problem is that the obtained sequence is not always convergent. Newton’s method works sometimes only locally. When the involved function (or operator) is convex and isotone (monotone increasing) (or anti-isotone), with continuous derivative of first order, the method leads to a rapidly convergent sequence. Its limit is the zero of the given function (respectively operator), which can be approximated with the control of the norm of the error. The strength of this method consists in the fact that is global, and the weakness is its limitation to convex (or concave) decreasing or increasing operators. The connection with the contraction principle is emphasized and proved in detail. In some cases, approximation in terms of a contraction constant is preferable to that from the classical Newton method. Chapter 8 is devoted to a special kind of functional equations. Both real and complex cases are under attention. Matrix equations are also briefly discussed. To conclude, old results have been recently applied to obtain new theorems, published in [39], [52] [53], [55], [58-63]. Some other results on polynomial approximation on unbounded subsets were motivated by solving Markov moment problems.

CHAPTER ONE ON HAHN-BANACH TYPE THEOREMS, POLYNOMIAL APPROXIMATION ON UNBOUNDED SUBSETS, THE MOMENT PROBLEM AND MAZUR-ORLICZ THEOREM

1. Introduction We recall the classical formulation of the moment problem, under the terms of T. Stieltjes, given in 1894-1895 (see the basic book of N.I. Akhiezer [1] for details): find the repartition of the positive mass on the nonnegative semi-axis, if the moments of arbitrary orders ݇ (݇ = 0,1,2, …) are given. Precisely, in the Stieltjes moment problem, a sequence of real numbers (ݏ )ஹ is given and one looks for a nondecreasing real function ߪ( ݐ( )ݐ 0), which verifies the moment conditions:

න ݐ ݀ߪ = ݏ ,

(݇ = 0,1,2, … ).

This is a one-dimensional moment problem, on an unbounded interval. Namely, is an interpolation problem with the constraint on the positivity of the measure dı. The numbers ݏ , ݇ אԳ = {0,1,2, … } are called the moments of the measure ݀ߪ. Existence, uniqueness, and construction of the solution ߪ are studied. The present work concerns firstly the existence problem. However, the uniqueness is studied as well. The moment problem is an inverse problem: we are looking for an unknown measure, starting from its moments. The direct problem might be: being given the measure ݀ߪ, compute its moments ݐ ݀ߪ, ݇ = 0,1,2, …. The connection with the positive polynomials and extensions of linear positive functional and operators is quite clear. Namely, if one denotes by ߮ , ߮ ( ݐ ؔ )ݐ , ݆ א Գ, [ א ݐ0, ), ࣪ the vector space of polynomials with real coefficients and

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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ܶ : ࣪ ՜ Թ, ܶ ቌ ߙ ߮ ቍ ؔ ߙ ݏ , אబ

אబ

where ܬ ؿԳ is a finite subset, then the moment conditions ܶ ൫߮ ൯ = ݏ , ݆ א Գ are clearly verified. It remains to check whether the linear form ܶ has nonnegative values at all nonnegative polynomials. If the latter condition is also accomplished, then one looks for the existence of a linear positive extension ܶ of ܶ to a larger ordered function space ܺ which contains both ࣪ and the space of continuous compactly supported functions, then representing ܶ by means of a positive regular Borel measure ߤ on [0, ), via Riesz representation theorem. Alternately one can apply directly Haviland theorem [25]. We start reviewing existence of a solution for simplest classical one-dimensional moment problems: the Hamburger moment problem (when the closed subset ك ܨԹ is = ܨԹ), Stieltjes moment problem (when = ܨԹା ) and Hausdorff moment problem (when [ = ܨ0,1]). In the sequel, the following notations are used: Գ = {0,1,2, … }, Թା = [0, λ), ܥ ( )ܨis the vector space of all real valued compactly supported continuous functions defined on ܨ, (ܥ ())ܨା is the convex cone of all functions in ܥ ( )ܨwhich take nonnegative values at each point of ܨ. ࣪ା ( )ܨis the convex cone of all polynomial functions with real coefficients, which are nonnegative on ܨ. If an interval (for example [ܽ, ܾ], Թ, or [0, λ)) is replaced by a closed subset ܨof Թ , ݊ 2, we have a multidimensional moment problem. Moments appear naturally in physics and probability theory, as discussed in the Introduction of [71]. Passing to an example of the multidimensional real classical moment problem, let us denote

߮ ( ݐ = )ݐ = ݐଵభ ݐ ڮ , ݆ = (݆ଵ , … , ݆ ) אԳ , ݐ( = ݐଵ , … , ݐ ) אԹା , ݊ אԳ, ݊ 2.

(1.1)

If a sequence ൫ݕ ൯אԳ is given, one studies the existence, uniqueness, and construction of a positive linear form ܶ defined on a function space containing polynomials and continuous real valued compactly supported functions, such that the moment conditions ܶ൫߮ ൯ = ݕ ,

݆ אԳ

(1.2)

6

Chapter One

are satisfied. Usually, the positive linear form ܶ can be represented by means of a positive regular Borel measure on Թା . When un upper constraint on the solution ܶ is required too, we have a Markov moment problem. This requirement is formulated as ܶ being dominated by a convex functional, which might be a norm, and its goal is to control the continuity and the norm of the solution. Clearly, the classical (Stieltjes) moment problem is an extension problem for a linear functional, from the subspace of polynomials to a function space which contains both polynomials as well as the continuous compactly supported real functions on Թା . From solutions linear functionals, many authors considered solutions linear operators. Of course, in this case the moments ݕ , ݆ אԳ are elements of an ordered vector space ܻ (usually ܻ is an order complete Banach lattice). The order completeness is necessary for applying Hahn-Banach type results for operators defined on polynomials and having ܻ as codomain. Various aspects of the classical moment problem have been studied (see the References). The case of multidimensional moment problem on compact semi-algebraic subsets in Թ was intensively studied. Clearly, the classical moment problem is related to the form of positive polynomials on the involved closed subsets of Թ . As it is well-known, there exist nonnegative polynomials on the entire space Թ , ݊ 2, which are not sums of squares of polynomials (see [4], [71]), contrary to the case ݊ = 1, which is reviewed in [1]. The analytic form of positive polynomials on special closed unbounded finite dimensional subsets is crucial in solving classical moment problems on such subsets (see [34] for the expression of nonnegative polynomials on a strip, in terms of sums of squares). Such results are useful in characterizing the existence of a positive solution by means of signatures of quadratic forms. In case of Markov moment problem, approximation of nonnegative compactly supported continuous functions (with their support contained in a closed subset) by special dominating polynomials on that subset, having known analytic form is very important. Based on the form of nonnegative polynomials on unbounded intervals and the above-mentioned approximation results, Markov moment problem on Cartesian products of unbounded intervals can be solved in terms of products of quadratic forms. In most of the cases, the uniqueness of the solution of the Markov moment problem on spaces ܮଵఔ ( )ܨfollows too, thanks to the density of polynomials in such spaces; here ܨis a closed unbounded subset of Թ and ߥ is a positive regular ܯ-determinate Borel measure on ܨ. Recall that a measure ߥ is ܯെdeterminate (moment determinate), if it has finite moments of all orders and is uniquely determined by its moments

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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න ݐ ݀ߥ, ݆ אԳ , ி

(or, equivalently, by its values on polynomials; see (1.1) for the notation ݐ , ݆ אԳ ). Connections of the moment problem with other fields, such as operator theory, fixed point theory, algebra, polynomial approximation, optimization, complex functions, are emphasized by means of the corresponding references. Applications of various moments in physics and probabilities theory are briefly reviewed in the Introduction of the monograph [71]. To conclude, for characterizing the existence of a solution of a classical moment problem in terms of moments, extension HahnBanach results and their generalizations accompanied by knowing the analytic form of positive polynomial on the set under discussion are the basic tools. Sometimes, especially in Markov moment problem, the uniqueness of the (continuous linear) solution follows too, by the proof of its existence, thanks to the density of polynomials in some function spaces (even on an unbounded closed subset). In this respect, for both existence and uniqueness of the solution, polynomial approximation on unbounded subsets is essential. Otherwise, the uniqueness problem requires specific methods (see [6], [71], [75], [76]). The rest of this chapter is organized as follows. Section 2 is devoted to recalling various results and methods on Hahn-Banach type extension for linear functionals and operators, preserving positivity or sandwich conditions, which are applied in the sequel. We start this discussion with geometric form of Hahn-Banach theorem, which leads rapidly to Krein-Milman theorem. In Section 3, earlier and recent results on existence and uniqueness of the solution for the classical moment problem are reviewed. Section 4 deals with moment problems and Mazur-Orlicz theorems in concrete spaces. Some of the solutions are Markov operators. Generally, the norms of the solutions can be determined by means of the norms of the bounded sublinear upper constraints. Section 5 is devoted to polynomial approximation on unbounded subsets and its applications to the Markov moment problem. As an application, the positivity of some continuous linear operators is characterized only in terms of quadratic forms (see Theorem 5. 13). From the point of view of this result, the difficulty created by the fact that on Թ , ݊ 2, there exist polynomials which are not sums of squares is solved.

8

Chapter One

2. Hahn-Banach type theorems, abstract Markov moment problems and Mazur-Orlicz theorem in ordered vector spaces setting Almost all the results of this section will be applied in the sequel. Some of them have been published first in [45], using earlier results on extension of linear operators preserving two constraints published in [42], [44]. Detailed proofs can be found in [44]. Before going on with the above-mentioned results, which are valid for operators, we review the geometric form of Hahn-Banach theorem. This variant is useful for some of the results stated in the next chapters. The following lemma is the key result for the direct proof of the geometric version of Hahn-Banach theorem (see [69], p. 4546). Lemma 2.1. (See [69]). Let ܺ be a real topological vector space (t.v.s.) of dimension at least 2. If ܦis an open convex subset not containing , there exists a one-dimensional subspace of ܺ not intersecting ܦ. Lemma 2.1 and a standard application of Zorn’s lemma yield the next result. Theorem 2.2. (See [69]). Let ܺ be a real t.v.s., let ܯbe a linear manifold in ܺ, and let ܦbe a non-empty open convex subset of ܺ, not intersecting ܯ. Then there exists a closed hyperplane ܪin ܺ, containing ܯand not intersecting ܦ. Corollary 2.3. Let ܧbe a t.v.s., ܥan open convex subset of ܧ, ܧଵ a vector subspace of ܧsuch that ܧଵ ് ܥ ת, ܶଵ ܧ(ܮ אଵ , Թ) a continuous linear functional, ܲ: ܥ՜ Թ a convex upper semi-continuous functional such that ܶଵ ( )ݔ ܲ( )ݔfor all ܧ א ݔଵ ܥ ת. Then there exists a continuous linear functional ܶ ܧ(ܮ א, Թ) which extends ܶଵ , such that ܶ( )ݔ ܲ( )ݔfor all א ݔ ܥ. To deduce Corollary 2.3 from Theorem 2.2 one applies the latter statement, where ܺ stands for × ܧԹ, ܯstands for the graph of ܶଵ ൫= ܯ ൛൫ݔ, ܶଵ ()ݔ൯; ܧ א ݔଵ ൟ൯, ܦstands for {(ݔ, × ܥ א )ݐԹ; ܲ(}ݐ < )ݔ. According to Theorem 2.2, there exists a closed hyperplane ܪin × ܧԹ which contains ܯ, such that = ܦ ת ܪ. Due to condition ܧଵ ് ܥ ת, ܪcannot be vertical, hence is the graph of a linear functional ܶ ܧ(ܮ א, Թ). From the details of this sketch of the proof, it is easy to observe that ܶ extends ܶଵ , ܶ( )ݔ ܲ()ݔ, ܥ א ݔand ܶ is continuous (and linear) from ܧto Թ (see also [69, Exercise 6 , p. 69]).

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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The next result holds in locally convex spaces. All such spaces are assumed to be Hausdorff. Theorem 2.4. (See [69, Theorem 4.2, p. 49]). Let ܺ be a t.v.s., whose topology is locally convex. If ܶଵ is a linear form, defined and continuous on a subspace ܯof ܺ, then ܶଵ has a continuous extension ܶ to the entire space ܺ. Corollary 2.5. Given ݊ { א1,2, … } and ݊ linearly independent elements ݔఔ of a l.c.s. ܺ, there exist ݊ continuous linear forms ܶఓ on ܺ such that ܶఓ (ݔఔ ) = ߜఓఔ , (ߤ, ߥ = 1, … , ݊). The next result is basic in finite dimensional convex analysis due to its applications, including the maximum principle for convex functions. Theorem 2.6. (Carathéodory; see [38] and/or [65]). Let ؿ ܭԹ (݊ { א1,2, … }) be a convex compact subset. Then any ܭ א ݔcan be written as convex combination of at most ݊ + 1 extreme points of ܭ. A simple proof of Theorem 2.6 (by induction on the dimension ݊) is given in [65, p. 7-8], essentially using Theorem 2.2 stated above. Here is a main application of Theorem 2.6 to convex optimization (in particular, to linear optimization). Corollary 2.7. (See [69, Exercise 26, p. 71]). Let ؿ ܭԹ be a nonempty compact subset. Then its convex hull ܿ )ܭ(is compact. Theorem 2.8. (See [38, p. 171). If ݂ is a continuous convex real function on a convex compact subset ؿ ܭԹ (݊ { א1,2, … }), then ݂ attains a global maximum at an extreme point of ܭ. Theorem 2.9. (The maximum principle [38]). Let ܥbe a convex subset of Թ . If a convex function ݂: ܥ՜ Թ attains its maximum on ܥat a point from the relative interior of ܥ, then ݂ is constant on ܥ. Next, we recall the following basic results, derived from Theorem 2.2. Theorem 2.10. (First Separation Theorem [69]). Let ܣbe a convex subset of a t.v.s. ܺ, such that ݅݊ ് )ܣ(ݐand let ܤbe a nonempty convex subset of ܺ, not intersecting the interior ݅݊ )ܣ(ݐof ܣ. There exists a closed hyperplane ܪseparating ܣand ;ܤif ܣand ܤare both open, ܪseparates ܣ and ܤstrictly.

10

Chapter One

Theorem 2.11. (Second Separation Theorem [69]). Let ܣ, ܤbe nonempty, disjoint convex subsets of a locally convex Hausdorff space (l.c.s.) ܺ, such that ܣis closed and ܤis compact. There exists a closed hyperplane in ܺ strictly separating ܣand ܤ. Corollary 2.12. Let ܺ be a l.c.s. and ݔଵ , ݔଶ ܺ א, ݔଵ ് ݔଶ . Then there exists a continuous linear functional څ ܺ א څ ݔsuch that ݔ( څ ݔଵ ) ് ݔ( څ ݔଶ ). The preceding corollary says that the topological dual ܺ څof a l.c.s. ܺ separates the points of ܺ. On the other hand, by the definition of weak topology on a l.c.s. ܺ, any weak closed subset of ܺ is closed in the initial topology on ܺ. For convex closed subsets, the reverse implication holds as well. Namely, we recall the following well-known consequence of Theorem 2.11: Corollary 2.13. (See [69]) Let ܺ be a locally convex space and ܺ ؿ ܥa convex closed subset. Then ܥis the intersection of all closed half-spaces containing it. In particular, ܥis closed with respect to the weak topology ܺ(ݓ, ܺ ) څon ܺ. The next key lemma is used in the proof of the main Theorem 2.15 (KreinMilman). Lemma 2.14. (See [69]). If ܥis a compact, convex subset of a locally convex space, every closed hyperplane supporting ܥcontains at least one extreme point of ܥ. We recall that, by definition a closed hyperplane ܪin the locally convex space ܺ under attention is supporting ܥif ് ܪ ת ܥand ܥis contained in one of the two half-spaces defined by ܪ. A point ݁ ܥ אis called an extreme point of ܥif from ݔଵ , ݔଶ ܥ א, ( א ݐ0,1), the equality ݁ = (1 െ ݔ)ݐଵ + ݔݐଶ implies ݔଵ = ݔଶ = ݁. In other words, ݁ cannot be an interior element of any line segment of ends elements of ܥ. Theorem 2.15. (Krein-Milman; see [69]). Every compact convex subset of a locally convex space is the closed convex hull of its extreme points. Krein-Milman theorem says that in any compact convex subset ܥof a l.c.s. there are many extreme points, which generate ( ܥany element of ܥis the limit of a net whose elements are convex combinations of extreme points of )ܥ.

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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Theorem 2.16. (See [69, Theorem 10.5, p. 68]). If ܭis a compact subset of a locally convex space such that the closed convex hull ܥof ܭis compact, then each extreme point of ܥis an element of ܭ. From Theorem 2.6 (Carathéodory), Corollary 2.7 and Theorem 2.16, the following consequence follows: Corollary 2.17. If ؿ ܭԹ is a compact nonempty subset, then its convex hull ܿ )ܭ(is compact and ܿ))ܭ(ݎݐݔܧ(ܿ = )ܭ(. Moreover, each point of ܿ )ܭ(can be written as convex combination of at most ݊ + 1 extreme points of ܭ. The above results are deduced from the geometric form of Hahn-Banach theorem. In most of cases, motivated by further applications, analytic proofs of Hahn-Banach type theorems are more suitable. Here is a first main result, completely proved in [38, p. 339-340]. Theorem 2.18. (The Hahn-Banach theorem). Let ܺ be a vector space, ܲ: ܺ ՜ Թ a sublinear functional, ܺ ؿ ܯa vector subspace ܮ: ܯ՜ Թ a linear functional, such that )ݔ(ܮ ܲ( )ݔfor all ܯ א ݔ. Then ܮhas a linear extension ܶ: ܺ ՜ Թ, such that ܶ is dominated by ܲ on the entire space ܺ. Corollary 2.19. (See [38, p. 340]) If ܲ is a sublinear functional on a real vector space ܺ, then for every element ݔ ܺ אthere exists a linear functional ܶ such that ܶ(ݔ ) = ܲ(ݔ ) and ܶ( )ݔ ܲ( )ݔfor all ܺ א ݔ. Theorem 2.20. (The Hahn-Banach theorem on normed vector spaces; see [38]). Let ܺ be a vector subspace of the real normed vector space ܺ and ܶ : ܺ ՜ Թ a continuous linear functional. Then ܶ has a continuous linear extension ܶ: ܺ ՜ Թ, with ԡܶԡ = ԡܶ ԡ. Corollary 2.21. (See [38]). If ܺ is normed vector space, then for each ݔ א ܺ, ݔ ് , there exists a linear functional ܶ on ܺ, such that ܶ(ݔ ) = ԡݔ ԡ, and ԡܶԡ = 1. One of the reasons of using analytic proofs of Hahn-Banach type theorems is that they work not only for extending linear functionals, but also for operators. As in the case of functional, the proofs of such type results are quite simple, by means of Zorn’s Lemma and extension of linear operators from a subspace ܵ of the involved domain space ܺ, to a space ܵ ْ

12

Chapter One

ܵݔ{݊ܽ }, where ݔ ܵ\ܺ א, preserving some constraints on the extension. The codomains of the operators for which Hahn-Banach type theorems hold must be order complete vector spaces, or even order complete vector lattices. We recall that an ordered vector space is a vector space ܻ endowed with an order relation which is compatible with the algebraic structure of a vector space. Namely, the following two properties are satisfied: ݕଵ ݕଶ , ݕ ฺ ܻ א ݕଵ + ݕ ݕଶ + ݕ, ݕଵ ݕଶ , ߙ אԹା ฺ ߙݕଵ ߙݕଶ . We say that such an order relation is linear. If ܻ is an ordered vector space, then ܻା = { ݕ ;ܻ א ݕ } is a convex cone, called the positive cone of ܻ. We always assume that the positive cone is generating (ܻ = ܻା െ ܻା ). An ordered vector space ܻ is called order complete (Dedekind complete) if for any upper bounded subset ܻ ؿ ܤ, there exists a least upper bound for ܤin ܻ, denoted by )ܤ(ݑݏ. A vector lattice is an ordered vector space ܻ with the property that for any ݕଵ , ݕଶ ܻ א, there exists ݕ{ݑݏଵ , ݕଶ } ܻ א. In a vector lattice ܻ, for any element ܻ א ݕone denotes |ݕ{ݑݏ = |ݕ, െ}ݕ. An ordered Banach space is a Banach space ܻ which is also an ordered vector space, such that the positive cone ܻା is closed and the norm is monotone on ܻା : 0 ݕଵ ݕଶ ฺ ԡݕଵ ԡ ԡݕଶ ԡ. A Banach lattice ܻ is a Banach space, which is also a vector lattice, such that ݕଵ , ݕଶ ܻ א, |ݕଵ | |ݕଶ | ฺ ԡݕଵ ԡ ԡݕଶ ԡ. Obviously, any Banach lattice is an ordered Banach space. In an ordered Banach space, there exists also a compatibility of the topology defined by the norm with the order relation. There exist ordered Banach spaces which are not lattices. For example, the space ܻ of all ݊ × ݊ symmetric matrixes with real entries, endowed with the norm ԡܸԡ = max |< ܸݔ, |> ݔ ԡ௫ԡஸଵ

and the order relation ܸ ܹ ع < ܸݔ, > ݔ < ܹݔ, > ݔfor all א ݔ Թ , ܸ, ܹ ܻ א, is an ordered Banach space which is not a lattice for ݊ 2. Here the norm ԡݔԡ is the Euclidean norm of the vector א ݔԹ . In the same way, if ܪis a real or complex Hilbert space, the real vector space ܻ = ࣛ()ܪ of all self-adjoint operators acting on ܪ, with the norm and order relation defined similarly to the case of symmetric matrixes, is an ordered Banach space which is not a lattice (here Թ is replaced by )ܪ. Almost all usual

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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function spaces and sequence spaces have natural structures of Banach lattices. On a vector space ࣠(ܵ) of real valued functions defined on a set ܵ, the usual order relation is: ݂ ݃ ݂( )ݐ ݃( )ݐfor all ܵ א ݐ. For example, if ܵ is a compact Hausdorff topological space, the space )ܵ(ܥof all real valued continuous functions over ܵ, is a Banach lattice with respect to the above defined order relation and usual norm. If we additionally assume that ܵ is connected and contains at least two different points, then )ܵ(ܥis not order complete. A particular such a Banach lattice is [(ܥ0,1]). The Lebesgue spaces ܮ ()ܨ, 1 λ, ك ܨԹ , and the sequence spaces ݈ , 1 , are order complete Banach lattices. Here is one of the old results on this subject, with many applications to the vector valued moment problem. Let ܺଵ be an ordered vector space whose positive cone ܺଵ,ା is generating (ܺଵ = ܺଵ,ା െ ܺଵ,ା ). Recall that in such an ordered vector space ܺଵ a vector subspace ܵ is called a majorizing subspace if for any ܺ א ݔଵ there exists ܵ א ݏsuch that ݔ ݏ. The following theorem holds true. Theorem 2.22. (See [30, Theorem 1.2.1]). Let ܺଵ be an ordered vector space whose positive cone is generating, ܺ ܺ ؿଵ a majorizing vector subspace, ܻ an order complete vector space, ܶ : ܺ ՜ ܻ a positive linear operator. Then ܶ admits a positive linear extension ܶ: ܺଵ ՜ ܻ. We go on with Hahn-Banach type theorems. Now a condition on the operator solution of being dominated by a convex operator defined on a convex subset of the domain space is required. In other words, a generalized Hahn-Banach theorem will be reviewed. The relationship between the next result and its corollary (existence of subgradients of convex operators) will appear clearly. A point ݔ of the subset ܣof a vector space ܺ is called an (algebraic) interior point of ܣif for each ܺ א ݔthere is a positive ߣ such that ߣ ݔ+ (1 െ ߣ)ݔ ܣ אfor |ߣ| ߣ . The point ݔ is said to be an (algebraic) relative interior point of ܣif for each ݔof the affine variety generated by ( ܣaffine hull of )ܣthere is a positive ߣ such that ߣ ݔ+ (1 െ ߣ)ݔ ܣ אfor |ߣ| ߣ . The set of all interior points of ܣis denoted by ܣ௧ and the set of all relative interior points by ܣ . For the next result see [81, Theorem 2.1, p. 284-286]. Theorem 2.23. (A generalized Hahn-Banach theorem; see [81]). Let ܺ be a vector space, ܺ ؿ ܯa vector subspace, ܻ an order complete vector space, ܺ ك ܣa convex subset, ܲ: ܣ՜ ܻ a convex operator, ܶெ : ܯ՜ ܻ a linear operator such that

14

Chapter One

ܶெ ( )ݔ ܲ( )ݔfor all ܣ ת ܯ א ݔ. If ܣ௧ ് ܯ ת, then there exists a linear operator ܶ: ܺ ՜ ܻ such that ܶ(ܶ = )ݔெ ( )ݔfor all ܯ א ݔand ܶ( )ݔ ܲ( )ݔfor all ܣ א ݔ. Corollary 2.24. (See [81, Corollary 2.7, p. 286]). Let ܺ be a vector space, ܻ an order complete vector space, ܺ ك ܣa convex subset, ܲ: ܣ՜ ܻ a convex operator. If ݔ ܣ א , then there exists a linear operator ܶ: ܺ ՜ ܻ such that ܶ( )ݔെ ܶ(ݔ ) ܲ( )ݔെ ܲ(ݔ ) for all ܣ א ݔ.

(1.3)

A linear operator ܶ satisfying (1.3) is called a subgradient of ܲ at ݔ . Corollary 2.24 says that a convex operator having as codomain an order complete vector space possesses a subgradient at every relative interior point of its domain. This result (with a somewhat different proof) goes back to [77]. The set of all subgradients of ܲ at ݔ is called the subdifferential of ܲ at ݔ and is denoted by ߲௫బ ܲ. This is a convex set, and, for convex operators ܲ satisfying the hypothesis of Corollary 2.24, is nonempty. In the results stated above, the order relation who naturally exists on concrete spaces does not appear on the domain space ܺ in any way. The next results take into consideration linear order structures on ܺ as well. This way, from now on, we have three conditions on the linear operator solution ܶ. Namely ܶ must extend a given linear operator defined on a subspace of ܺ, it is dominated by a given convex operator ܲ and dominates a given concave operator ܳ. If ܳ|శ , then the linear extension ܶ is positive: א ݔ ܺା ฺ ܶ(ܻ א )ݔା . Recall that an ordered vector space ܺ which is also a topological vector space is called an ordered topological vector space if the positive cone ܺା is topologically closed. The next result is due to H. Bauer and independently to I. Namioka (for citation of the original sources see [69, p. 227]). Theorem 2.25. (See [69, Theorem 5.4, p. 227]). Let ܺ be an ordered t.v.s. with positive cone ܺା and ܯa vector subspace of ܺ. For a linear form ܶ on ܯto have a linear continuous positive extension ܶ: ܺ ՜ Թ it is necessary and sufficient that ܶ be bounded above on ܷ( ת ܯെ ܺା ), where ܷ is a suitable convex െneighborhood in ܺ. The next result is motivated by Theorem 2.25 and the discussion preceding it. In the sequel, all theorems are valid for operators. In particular, the

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

15

corresponding cases of real valued functionals follow as consequences. In the next theorem, ܺ will be a real vector space, ܻ an order-complete vector lattice, ܣ, ܺ ك ܤconvex subsets, ܳ: ܣ՜ ܻ a concave operator, ܲ: ܤ՜ ܻ a convex operator, ܺ ؿ ܯa vector subspace, ܶ : ܯ՜ ܻ a linear operator. All vector spaces and linear operators are considered over the real field. Theorem 2.26. (See [42, Theorem 1]). Assume that ܶ ( )ݔ ܳ(א ݔ )ݔ ܣ ת ܯ, ܶ ( )ݔ ܲ(ܤ ת ܯ א ݔ )ݔ. The following two statements are equivalent: (a) there exists a linear extension ܶ: ܺ ՜ ܻ of the operator ܶ such that ܶ| ܳ, ܶ| ܲ; (b) there exists ܲଵ : ܣ՜ ܻ, convex, and ܳଵ : ܤ՜ ܻ concave operator such that for all (ߩ, ݐ, ߣ, ܽଵ , ܽ, ܾଵ , ܾ, [ א )ݒ0,1]ଶ × (0, λ) × ܣଶ × ܤଶ × ܯ, the following implication holds: (1 െ ܽ)ݐଵ െ ܾݐଵ = ݒ+ ߣ൫(1 െ ߩ)ܽ െ ߩܾ൯ ฺ (1 െ ܲ)ݐଵ (ܽଵ ) െ ܳݐଵ (ܾଵ ) ܶ ( )ݒ+ ߣ൫(1 െ ߩ)ܳ(ܽ) െ ߩܲ(ܾ)൯. Notable, the extension ܶ of Theorem 2.26 satisfies the following conditions: is an extension of ܶ , it is dominated by ܲ on ܤand it dominates ܳ on ܣ. Here the convex subsets ܣ, ܤare arbitrary, with no restriction on the existence of relative interior points or on their position with respect to the subspace ܯ. The following theorems follow more or less directly as corollaries of Theorem 2.26. For details and applications to the abstract Markov moment problem see [42], [44], [45]. For applications to characterizing the isotonicity of a convex operator over a convex cone see [39]. The same article [39] contains a large class of examples of concrete spaces and operators for which the developed theory works. From Theorem 2.26 we obtain. Theorem 2.27. (See [42], [44]). Let ܺ be an ordered vector space, ܻ an order complete vector space, ܺ ؿ ܯa vector subspace, ܶଵ : ܯ՜ ܻ a linear operator, ܲ: ܺ ՜ ܻ a convex operator. The following statements are equivalent:

16

Chapter One

(a) there exists a positive linear extension ܶ: ܺ ՜ ܻ of ܶଵ such that ܶ ܲ on ܺ; (b) we have ܶଵ (݄) ܲ( )ݔfor all (݄, ܺ × ܯ א )ݔsuch that ݄ ݔ. One observes that in the very particular case ܧା = {}, when the order relation on ܧis the equality, from Theorem 2.27 one obtains the HahnBanach extension theorem for linear operators dominated by convex operators stated below. Corollary 2.28. Let ܺ be a vector space, ܻ an order complete vector space, ܺ ؿ ܯa vector subspace, ܶଵ : ܯ՜ ܻ a linear operator, ܲ: ܺ ՜ ܻ a convex operator. Assume that ܶଵ ( )ݔ ܲ( )ݔfor all ܯ א ݔ. Then there exists a linear extension ܶ: ܺ ՜ ܻ of ܶଵ such that ܶ( )ݔ ܲ( )ݔfor all ܺ א ݔ. Theorem 2.27 is equivalent to the following result, formulated in the abstract Markov moment setting. Theorem 2.29. (See [45, Theorem 1]). Let ܺ be a preordered vector space, ܻ an order complete vector lattice, ܲ: ܺ ՜ ܻ a convex operator, ൛ݔ ൟא ؿ ܺ, ൛ݕ ൟא ܻ ؿgiven families. The following statements are equivalent: (a) there exists a linear positive operator ܶ: ܺ ՜ ܻ such that ܶ൫ݔ ൯ = ݕ , ݆ ܬ א, ܶ( )ݔ ܲ()ݔ, ;ܺ א ݔ (b) for any finite subset ܬ ܬ ك, ܽ݊݀ ܽ݊ ݕ൛ߣ ; ݆ ܬ א ൟ كԹ, , we have ߣ ݔ ฺ ܺ א ݔ ߣ ݕ ܲ()ݔ. אబ

אబ

If we additionally assume that ܲ is isotone ൫ ݑ )ݑ(ܲ ֜ ݒ ܲ()ݒ൯, the assertions (a) and (b) are equivalent to (c), where: (c) for any finite subset ܬ ܬ ؿand any ൛ߣ ൟא ؿԹ, the following బ

inequality holds: ߣ ݕ ܲ ቌ ߣ ݔ ቍ. אబ

אబ

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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When the convex operator ܲ is defined only on the positive cone of ܺ, one obtains the following variant of Theorem 2.27 (see [44]): Theorem 2.30. (See [44], [59]). Let ܺ be an ordered vector space, ܻ an order complete vector space, ܺ ؿ ܯa vector subspace, ܶଵ : ܯ՜ ܻ a linear operator, ܲ: ܺା ՜ ܻ a convex operator. The following statements are equivalent: (a) there exists a positive linear extension ܶ: ܺ ՜ ܻ of ܶଵ such that ܶ|శ ܲ; (b) we have ܶଵ (݄) ܲ( )ݔfor all (݄, ܺ × ܯ א )ݔା such that ݄ ݔ. Proof. The implication (ܽ) ฺ (ܾ) is obvious; indeed, we have ܶଵ (݄) = ܶ(݄) ܶ( )ݔ ܲ()ݔ, for all (݄, ܺ × ܯ א )ݔା such that ݄ ݔ, thanks to the positivity of ܶ, also using the property ܶ( )ݔ ܲ( )ݔfor all ܧ א ݔା . To prove the converse, let ܲ be an arbitrary convex operator over ܺା , verifying the conditions mentioned at (b). We are going to apply Zorn lemma to the set ࣦ of all pairs (ܭ, ܶ ), where ܭis a vector subspace of ܺ, ܭ ؿ ܯ, ܶ : ܭ՜ ܻ is a linear operator such that ܶ |ு = ܶଵ and ܶ (݄) ܲ( )ݔfor all ݄ ܭ אand ܺ א ݔା such that ݄ ݔ. The set ࣦ contains the pair (ܯ, ܶଵ ) and is inductively ordered by the order relation (ܭ, ܶ ) ܼ( ا, ܶ ) ֞ ܼ ؿ ܭ, ܶ | = ܶ . According to Zorn’s Lemma, there exists a maximal pair ൫ܪெ , ܶுಾ ൯ ࣦ א. Our aim is to prove that ܪெ = ܺ. Assuming this is done, and taking ܶ ؔ ܶுಾ , we have ܶ(݄) ܲ( )ݔfor all ݄ ܯ אand ܧ א ݔା such that ݄ ݔ. Application of this inequality for ݄ = െ݊ݕ, ܺ א ݕା , ݊ אԳ, = ݔ0, yields ݊ܶ(െ(ܶ = )ݕെ݊ )ݕ ܲ(), ݊ אԳ. Since any order complete vector space is Archimedean, it results ܶ(െ )ݕ , ܺ א ݕା , that is the positivity of ܶ. Also, taking ݄ = ܺ א ݔା , one obtains ܶ( )ݔ ܲ( )ݔand ܶ will be the expected positive extension of ܶଵ , ܶ|శ ܲ. This will end the proof. If ܪெ ് ܺ, then we can choose ݒ ܪ\ܺ אெ and define ܪ = ܪெ ۩Թݒ , ܶ : ܪ ՜ ܻ, ܶ (݄ + ݒݎ ) = ܶுಾ (݄) + ݕݎ , ݄ ܪ אெ , א ݎԹ,

18

Chapter One

where ݕ ܻ אwill be chosen such that ൫ܪெ , ܶுಾ ൯ ܪ( ا , ܶ ) and (ܪ , ܶ ) ࣦ א. This will contradict the maximality of ൫ܪெ , ܶுಾ ൯ in ࣦ. Thus ܪெ = ܺ. To prove that (ܪ , ܶ ) ࣦ אfor suitable ݕ ܻ א, we have to show that ݄ ܪ אெ , א ݎԹ, ݄ + ݒݎ ܺ א ݔା ֜ ܶுಾ (݄) + ݕݎ ܲ()ݔ. For > ߙ = ݎ0, multiplying by ߙ ିଵ the relation ݄ଵ + ߙݒ ݔଵ ܺ אା , the above implication becomes: ݄ଵ + ߙݒ ݔଵ ܺ אା ֜ ݕ ߙ ିଵ ቀܲ(ݔଵ ) െ ܶுಾ (݄ଵ )ቁ , ߙ > 0, and, similarly, ݄ଶ + ߚݒ ݔଶ ܺ אା ֜ ݕ ߚ ିଵ ቀܲ(ݔଶ ) െ ܶுಾ (݄ଶ )ቁ , ߚ < 0. To have both conditions on ݕ verified, according to order completeness of ܻ, it is necessary and sufficient to prove that ߚ ିଵ ቀܲ(ݔଶ ) െ ܶுಾ (݄ଶ )ቁ ߙ ିଵ ቀܲ(ݔଵ ) െ ܶுಾ (݄ଵ )ቁ. The last inequality may be written as ܶுಾ (ߙ ିଵ ݄ଵ െ ߚ ିଵ ݄ଶ ) ߙ ିଵ ܲ(ݔଵ ) െ ߚ ିଵ ܲ(ݔଶ ) To prove this last inequality, we eliminate ݒ by adding the previous inequalities, multiplied by ߙ ିଵ > 0, respectively by െߚ ିଵ > 0, as follows: (ߙ ିଵ ݄ଵ + ݒ ߙ ିଵ ݔଵ , െߚ ିଵ ݄ଶ െ ݒ െߚ ିଵ ݔଶ ) ֜ ߙ ିଵ ݄ଵ െ ߚ ିଵ ݄ଶ ߙ ିଵ ݔଵ െߚ ିଵ ݔଶ ܺ אା , where ݄ ܪ אெ , ݔ ܺ אା , ݆ = 1,2, ߙ > 0, ߚ < 0. Since ൫ܪெ , ܶுಾ ൯ ࣦ אand ܲ is convex, these further yields: 1 ܶ (ߙ ିଵ ݄ଵ െ ߚ ିଵ ݄ଶ ) െ ߚ ିଵ ுಾ ߙ ିଵ െߚ ିଵ ݄ + ݄ ቇ = ܶுಾ ቆ ିଵ ଵ ߙ െ ߚ ିଵ ߙ ିଵ െ ߚ ିଵ ଶ

ߙ ିଵ

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

ܲቆ

19

ߙ ିଵ െߚ ିଵ ݔଵ + ିଵ ݔቇ ିଵ െߚ ߙ െ ߚ ିଵ ଶ

ߙ ିଵ

െߚ ିଵ ߙ ିଵ ) + ܲ(ݔ ܲ(ݔଶ ). ଵ ߙ ିଵ െ ߚ ିଵ ߙ ିଵ െ ߚ ିଵ Thus, the expected inequality follows, DQGWKHSURRILVFRPSOHWHƑ The next result provides a sufficient condition on the given linear operators for the existence of the linear extensions. When ܺ = Թଶ , ܻ = Թ, it has an interesting geometric meaning. Theorem 2.31. (See [44]). Let ܺ be a locally convex space, ܻ an order complete vector lattice with strong order unit ݑ and ܵ ܺ ؿa vector subspace. Let ܺ ؿ ܣbe a convex subset with the following properties: (a) there exists a neighborhood ܸ of the origin such that (ܵ + ܸ) ( ; = ܣ תthat is, by definition, ܣand ܵ are distanced). (b) ܣis bounded. For any equicontinuous family of linear operators ൛݂ ൟא ܵ(ࣦ ؿ, ܻ) and for any ݕ ܻ אା \{}, there exists an equicontinuous family ൛ܶ ൟא ܺ(ࣦ ؿ, ܻ) such that ܶ (݂ = )ݏ ()ݏ, ܵ א ݏ,

ܶ (߰) ݕ, ߰ ܣ א, ݆ ܬ א.

Moreover, if ܸ is a convex balanced neighbourhood of the origin such that ݂ (V תS) [ ؿെݑ , ݑ ],

(ܵ + ܸ) = ܣ ת,

and if ߙ > 0 is such that ܲ (ܽ) ߙ ܣ א ܽand ߙଵ > 0 is large enough such that ݕ ߙଵ ݑ , then the following relations hold ܶ ( )ݔ (1 + ߙ + ߙଵ )ܲ (ݑ)ݔ ,

ܺ א ݔ, ݆ ܬ א.

We have denoted by ܲ the gauge attached to ܸ. The next result was published in the following version in [45, Theorem 4]. It can be regarded as a generalization of a result of M.G. Krein [29] to arbitrary infinite set of moment interpolation conditions and to vector

20

Chapter One

valued mappings. This result is formulated in terms of the abstract Markov moment problem. Theorem 2.32. (See [45]). Let ܺ be an ordered vector space, ܻ an order complete vector lattice, ൛߮ ൟא ܺ ؿ, ൛ݕ ൟא ܻ ؿgiven arbitrary families, ܶଵ , ܶଶ ܺ(ܮ א, ܻ) two linear operators. The following statements are equivalent: (a) there is a linear operator ܶ ܺ(ܮ א, ܻ) such that ܶଵ ( )ݔ ܶ( )ݔ ܶଶ (ܺ א ݔ )ݔା , ܶ൫߮ ൯ = ݕ ;ܬ א ݆ (b) for any finite subset ܬ ܬ ؿand any ൛ߣ ; ݆ ܬ א ൟ ؿԹ, the following implication holds true

ቌ ߣ ߮ = ߰ଶ െ ߰ଵ , ߰ଵ , ߰ଶ ܺ אା ቍ ฺ ߣ ݕ ܶଶ (߰ଶ ) െ ܶଵ (߰ଵ ). אబ

אబ

If ܺ is a vector lattice, then assertions (a) and (b) are equivalent to (c), where: (c) ܶଵ ( )ݓ ܶଶ ( )ݓfor all ܺ א ݓା and for any finite subset ܬ ܬ ؿand ൛ߣ ; ݆ ܬ א ൟ ؿԹ, we have ା

ି

ߣ ݕ ܶଶ ቌቌ ߣ ߮ ቍ ቍ െ ܶଵ ቌቌ ߣ ߮ ቍ ቍ אబ

אబ

אబ

The following theorem is also a Hahn-Banach- type result (see Theorem 2.29 stated above) but is formulated in terms which are similar to those of the abstract Markov moment problem [45]. However, the condition ܶ൫ݔ ൯ = ݕ , ݆ ܬ אof the abstract moment problem (Theorem 2.29 stated above) is replaced by ܶ൫ݔ ൯ ݕ , ݆ ܬ א. Consequently it is sufficient that the implication mentioned at point (b) holds only for nonnegative scalars ߣ . Theorem 2.33. (Mazur-Orlicz: see [45, Theorem 5]). Let ܺ be a preordered vector space, ܻ an order complete vector space, ൛ݔ ൟא , ൛ݕ ൟא families of elements in ܺ, respectively in ܻ, ܲ: ܺ ՜ ܻ a sublinear operator. The following statements are equivalent:

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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(a) there exists a linear positive operator ܶ: ܺ ՜ ܻ such that ܶ൫ݔ ൯ ݕ , ݆ ܬ א, ܶ( )ݔ ܲ()ݔ, ;ܺ א ݔ (b) for any finite subset ܬ ܬ ؿand any൛ߣ ൟא ؿԹା = [0, λ), the బ

following implication holds true ߣ ݔ ฺ ܺ א ݔ ߣ ݕ ܲ()ݔ. אబ

אబ

If in addition we assume that ܲ is isotone, the assertions (a) and (b) are equivalent to (c), where: (c) for any finite subset ܬ ܬ ؿand any ൛ߣ ൟא ؿԹା , the following బ

inequality holds

ߣ ݕ ܲ ቌ ߣ ݔ ቍ. אబ

אబ

In the end of this section, we state a general constrained extension result which can be proved as a consequence of Theorem 2.26. Probably Theorems 2.26 and 2.34 are equivalent. Theorem 2.34. (See [44]). Let ܺ be a vector space, ܻ be an order complete vector lattice, ܺ ؿ ܯa vector subspace, ܶ : ܯ՜ ܻ a linear operator, ك ܣ ܺ a convex subset, ܳ: ܣ՜ ܻ a concave operator. Assume that ܶ ( )ݔ ܳ(ܣ ת ܯ א ݔ )ݔ. The following statements are equivalent: (a) There exists a linear operator ܶ: ܺ ՜ ܻ which extends ܶ , such that ܶ| ܳ. (b) There exists a convex operator ܲ: ܣ՜ ܻ such that for all (ݔ, ݎ, ܽ) א ( × ܯ0, λ) × ܣthe following implication holds: ݔ+ ܶ ฺ ܣ א ܽݎ ( )ݔ+ )ܽ(ܳݎ ܲ( ݔ+ )ܽݎ. Moreover, if ܲ satisfies the requirements of (b), then the extension ܶ of (a) verifies the relation ܶ| ܲ. It seems that the general Theorems 2.26 and 2.34 are equivalent. On the other hand, since all concrete spaces are endowed with a natural linear order

22

Chapter One

relation, we restate Theorem 2.34 in the framework of ordered vector spaces. Theorem 2.35. Let ܺ be an ordered vector space, let ܻ be an order complete vector lattice, ܺ ؿ ܯa vector subspace, ܶ : ܯ՜ ܻ a linear operator, ܳ: ܺା ՜ ܻ a supralinear operator, ܲ: ܺା ՜ ܻ a convex operator. The following statements are equivalent: (a) There exists a linear operator ܶ: ܺ ՜ ܻ which extends ܶ , such that ܳ ܶ|శ ܲ. (b) For all (݄, ߮ଵ , ߮ଶ ) ܺ × ܯ אା × ܺା , the following implication holds: ݄ = ߮ଶ െ ߮ଵ ฺ ܶ (݄) ܲ(߮ଶ ) െ ܳ(߮ଵ ). The next consequence of Theorem 2.35 is a sandwich type result. It can be obtained from Theorem 2.35, applied to { = ܯ}, ܶ = . Corollary 2.36. Let ܺ, ܻ, ܲ, ܳ be as in the statement of Theorem 2.35. Assume that ܳ ܲ on ܺା . Then there exists a linear operator ܶ: ܺ ՜ ܻ, such that ܳ ܶ|శ ܲ. The next two variants of the same controlled regularity property of some linear operators are also consequence of Theorem 2.26. Recall that a linear operator ܶ is called regular if it can be written as a difference of two positive linear operators ܸ, ܹ: ܶ = ܸ െ ܹ. If ܸ is dominated by a given convex operator ܲ, we say that we have a controlled regularity for ܶ. This terminology is motivated by the fact that in the topological framework, ܲ is assumed to be continuous and ܸ ܲ on the entire domain space usually implies the continuity of ܸ. Sometimes, the norm of ܸ can be evaluated or determinate as well. Theorem 2.37. (See [44]). Suppose that ܺ is an ordered vector space, ܻ is an order complete vector lattice and ܲ: ܺା ՜ ܻ is a convex operator. Then for any linear operator ܶ: ܺ ՜ ܻ the following two statements are equivalent: (a) there exist positive linear operators ܸ, ܹ: ܺ ՜ ܻ such that ܶ = ܸ െ ܹ, ܸ|శ ܲ. (b) ܶ(ݔଵ ) ܲ(ݔଶ ) for all ݔଵ , ݔଶ in ܺ such that ݔଵ ݔଶ .

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Most of convex operators ܲ appearing in applications are defined on the entire domain space. Therefore, we recall the similar statement to that of Theorem 2.37, but for convex operators ܲ: ܺ ՜ ܻ. Theorem 2.38. (See [42], [44]). Assume that ܺ is an ordered vector space, ܻ is an order complete vector lattice and ܲ: ܺ ՜ ܻ is a convex operator. For any linear operator ܶ: ܺ ՜ ܻ the following two assertions are equivalent: (a) there exist two positive linear operators ܸ, ܹ: ܺ ՜ ܻ such that ܶ = ܸ െ ܹ, ܸ ܲ; (b) ܶ(ݔଵ ) ܲ(ݔଶ ) for all ݔଵ , ݔଶ in ܺ such that ݔଵ ݔଶ . For recent applications to Theorems 2.30, 2.37, 2.38 to the isotonicity of convex operators over convex cones see [39]. On the other hand, Theorem 2.32 was applied in [2] to a sandwich result on finite simplicial sets and found a topological version in [59].

3. The moment problem The moment problem is important in itself. Moreover, in time, it leaded to new results in Analysis and Functional Analysis. On the other hand, applications of the classical moments in physics, probability theory and statistics are well-known. We start by briefly recalling some basic known results on existence of a solution for the one-dimensional moment problem (see [71]). For the next two theorems, the explicit forms of nonnegative polynomials on Թ and respectively on Թା , accompanied by Haviland theorem are applied. We recall that ࣪ א ା (Թ) if and only if = ଵଶ + ଶଶ for some ଵ , ଶ אԹ[ ]ݐand ࣪ א ା (Թା ) if and only if = )ݐ(ଵଶ ( )ݐ+ ݐଶଶ ()ݐ, א ݐԹା , for some ଵ , ଶ אԹ[]ݐ. Theorem 3.1. (Hamburger’s theorem; see [71, Theorem 3.8, p. 63]). For a real sequence = (ݕ )אԳ the following statements are pairwise equivalent: (i) is a Hamburger moment sequence, that is, there is a nonnegative Radon measure ߤ on Թ such that ݐ ܮ אଵఓ (Թ), ݆ אԳ and න ݐ ݀ߤ(ݕ = )ݐ , ݆ אԳ. Թ

24

Chapter One

(ii) The sequence ܡis positive semidefinite, i. e. for all ݊ אԳ and ݔ , ݔଵ , … , ݔ אԹ we have

ݕା ݔ ݔ 0. ,ୀ

(iii) All Hankel matrixes ܪ () = ൫ݕା ൯,ୀ , ݊ אԳ are positive semidefinite. (iv) ܶ defined by ܶ ൫σאబ ߙ ߮ ൯ ؔ σאబ ߙ ݕ , ܬ ؿԳ is a finite subset, ߙ אԹ, is a positive linear functional on Թ[]ݐ, that is ܶ (ଶ ) 0 for all א Թ[]ݐ. (v) ܶ ( )ݍ 0 for all ࣪ א ݍା (Թ). Theorem 3.2. (See [71, p. 65]). For a real sequence ݕ( = ݕ )אԳ the following statements are pairwise equivalent: (i)

is a Stieltjes moment sequence, that is, there is a nonnegative Radon measure ߤ on [0, λ) such that ݐ ܮ אଵఓ (Թା ), ݆ אԳ and

න ݐ ݀ߤ(ݕ = )ݐ , ݆ אԳ.

(ii) For all ݊ אԳ and ݔ , ݔଵ , … , ݔ אԹ we have

ݕା ݔ ݔ 0,

ݕାାଵ ݔ ݔ 0.

,ୀ

,ୀ

(iii) All Hankel matrixes ൫ݕା ൯,ୀ , ൫ݕାାଵ ൯,ୀ , ݊ אԳ, are positive semidefinite. (iv) ܶ (ଶ ) 0 and ܶ ( ݍݐଶ ) 0 for , א ݍԹ[]ݐ. (v) ܶ ( )ݍ 0 for all ࣪ א ݍା (Թା ). Theorem 3.1 (respectively 3.2) give necessary and sufficient conditions for a sequence = (ݕ )אԳ of real numbers to be an Թ െ moment sequence (respectively an Թା െ moment sequence). Next, we go on with the corresponding problem on [0,1] (the Hausdorff moment problem).

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Theorem 3.3. (See [71, p. 66]). For a real sequence , the following statements are pairwise equivalent: (i) is a [0,1] െ moment sequence. (ii) ܶ ((1 െ )ݐ ݐ ) 0 for ݊. ݇ אԳ. ݊ (iii) σୀ(െ1) ቀ ݆ ቁ ݕା 0, for ݊. ݇ אԳ. We go on with the problem of determinacy. A moment sequence is called determinate if it has only one representing measure. If a sequence has a representing measure supported on a compact subset ܨ, then is determinate thanks to Weierstrass approximation theorem. A Hamburger moment sequence is determinate if it has a unique representing measure, while a Stieltjes moment sequence is called determinate if it has only one representing measure supported on [0, λ). The Carleman theorem contains a powerful sufficient condition for determinacy. Theorem 3.4. (See [71, Theorem 4.3, p. 80-81]). Suppose that = (ݕ )אԳ is a positive semidefinite sequence. (i) If satisfies Carleman condition ି

ଵ

ݕଶଶ = +λ, ୀଵ

then is a determinate Hamburger moment sequence. (ii) If in addition (ݕାଵ )אԳ is positive definite and ି

ݕ

ଵ ଶ

= +λ,

ୀଵ

then is a determinate Stieltjes moment sequence. The following theorem of Krein consists in a sufficient condition for indeterminacy (for measures given by densities). Theorem 3.5. (Krein condition; see [71, Theorem 4.14, p. 85-86]). Let ݂ be a nonnegative Borel function on Թ. Suppose that the measure ߤ defined by

26

Chapter One

݀ߤ = ݂( ݐ݀)ݐis a Radon measure on Թ, and has finite moments ݕ ؔ Թ ݐ ݀ߤ for all ݊ אԳ. If െln൫݂()ݔ൯ ݀ < ݔ+λ, 1 + ݔଶ

න Թ

then the moment sequence ݕ( = ݕ )אԳ. is ܯെindeterminate. In what follows, we give new checkable sufficient conditions on distributions of random variables that imply Carleman condition, ensuring determinacy. Consider two random variables ܸ~ Ȍ, ܸ with values in Թ, ܹ~ ܩ, ܹ with values in Թା . Assume that both Ȍ and ܩhave continuous derivatives and let ߰ = ݀ȲΤ݀ݐ, respectively ݃ = ݀ܩ/݀ ݐbe the corresponding densities. All moments of ܸ, ܹ are assumed to be finite. The symbol ա used below has the usual meaning of “monotone increasing”. Theorem 3.6. (See [76, Theorem 1, p. 498]; Hamburger case). Assume that the distribution ߰ of ܸ is symmetric on Թ and continuous and strictly positive outside an interval (െݐ , ݐ ), ݐ > 1, such that the following conditions hold: െ ln ߰()ݐ ݀ = ݐ+λ, ଶ |௧|ஹ௧బ ݐln(|)|ݐ

න

ି ୪୬ ట(௧) ୪୬ ௧

ա +λ as ݐ ݐ՜ +λ.

Under these conditions, ܸ~ Ȍ satisfies Carleman’s condition, and hence it is ܯെdeterminate. Theorem 3.7. (See [76, Theorem 2, p. 498]; Stieltjes case). Assume that the density ݃ of ܹ is continuous and strictly positive on [ܽ, λ) for some ܽ > 1 such that the following conditions hold:

න ି ୪୬ (௧) ௧

െ ln ݃( ݐଶ ) ݀ = ݐ+λ, ݐଶ ln ݐ ա +λ as ܽ ݐ՜ +λ.

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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Under these conditions, ܹ~ ܩsatisfies Carleman’s condition, and hence it is ܯെdeterminate. Example 3.1. The distribution function ܩhaving as density g(= )ݑ ݁(ݔെ)ݑ, א ݑԹା , satisfies the conditions of Theorem 3.7, hence it is ܯെ determinate. The problem of determinacy for the multidimensional moment problem on unbounded subsets is much more difficult than that for the one-dimensional case. However, important results on existence of a solution of the multidimensional moment problem on compact subsets and other closed subsets have been published (see [1], [2], [4], [5], [12], [13], [25], [67], [70], [71], [78]). In [34], the explicit form of nonnegative polynomial on a strip in terms of sums of squares was proved. This leads to a solution for the Markov moment problem on a strip. Going back to the existence problem for a solution, we consider the multidimensional case, which is much more complicated than the one-dimensional moment problem. The main reason is that the analytic form of nonnegative polynomials on closed subsets of Թ , ݊ 2, is generally not known in terms of sums of squares of polynomials. A case when this difficulty can be solved is that of semialgebraic compact subsets of Թ . If = ൫ݕ ൯אԳ ݊ 2, is a sequence of real numbers, one denotes by ܶ the linear functional defined on Թ[ݐଵ , … , ݐ ] by ܶ ቌ ߙ ݐ ቍ = ߙ ݕ , אబ

אబ

where ܬ ؿԳ is a finite subset and ߙ are arbitrary real coefficients. Let {݂ଵ , … , ݂ } be a finite subset of Թ[ݐଵ , … , ݐ ]. Then the closed subset given by א ݐ{ = ܭԹ ; ݂ଵ ( )ݐ 0, … , ݂ ( )ݐ 0}

(1.4)

is called a semi-algebraic set. The following result was proved for compact semi-algebraic sets. Many of the results of [12] hold true for other compact (non-semi-algebraic) subsets of Թ . Interesting aspects of reduced moment problem on unbounded intervals Թ, Թା are discussed in [27].

28

Chapter One

Theorem 3.8. (See [70], [71]). Let ܭbe a compact semi-algebraic set as defined by (1.4). Then there is a positive Borel measure ߤ supported on ܭ such that න ݐ ݀ߤ = ݕ , א ݆Գ ,

if and only if

ܶ ൫݂ଵ భ ݂ ڮ ೖ ଶ ൯ 0, א Թ[ݐଵ , … , ݐ ], ݁ଵ , … , ݁ { א0,1}. Corollary 3.9. (See [70]). With the above notations, if א Թ[ݐଵ , … , ݐ ] is such that > )ݐ(0 for all ݐin the semi-algebraic compact ܭdefined by (1.4), then is a finite sum of special polynomials of the form

݂ଵ భ ݂ ڮ ೖ ݍଶ 0, for some א ݍԹ[ݐଵ , … , ݐ ] and ݁ଵ , … , ݁ { א0,1}. Corollary 3.9 is named Schmüdgen’s Positivstellensatz. There also exists Putinar’s Positivstellensatz. These are representations of positive polynomials on basic closed semi-algebraic sets, in terms of sums of squares and polynomials defining the semi-algebraic set under attention. Interesting relationship between the Hausdorff moment problem and fixed-point results is emphasized in [7]. A Hausdorff moment sequence (݉ )ஹ is pointed out as a fixed point for an appropriate transformation from the convex set of all Hausdorff sequences associated to corresponding probability measures, to itself (such sequences are called normalized Hausdorff sequences). In [7], the properties of the fixed-point sequence are obtained, and the representing measure is determined by means of complex analysis techniques. The transformation ܶ under attention is defined by ܶ൫(ݕ )൯ =

1 , ݊ { א0,1,2, … . } 1 + ݕଵ + ݕ ڮ

(1.5)

It defines naturally a transformation from the set of all probability measures on [0,1] to itself. A fixed point (݉ )ஹ of ܶ verifies (according to (1.5)): ݉ = 1, (1 + ݉ଵ + ݉ ڮ )݉ = 1, ݊ 1.

(1.6)

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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From (1.6) we clearly infer the basic recurrence relation: ݉ାଵ + 1Τ݉ = 1Τ݉ାଵ , which can be written as ଶ + ݉ାଵ Τ݉ െ 1 = 0. ݉ାଵ

(1.7)

On the other hand, for any normalized Hausdorff (ݕ )ஹ we have 1 = ݕ ݕଵ ڮ ݕାଵ ڮ 0. For such a sequence, there exists lim ݕ = inf ݕ 0,

՜

and equality occurs in the last inequality if and only if the measure ߤ representing the sequence (ݕ )ஹ satisfies ߤ({1}) = 0. The Lebesgue measure ݀ ݐclearly has this property. From (1.7), we derive: lim ݉ = 0.

՜

Among many other main properties of the fixed sequence (݉ )ஹ , it was proved in [7] that ݉ ~ 1Τξ2݊ , ݊ ՜ λ. Remark. Let ك ܨԹ be an arbitrary closed subset, and Ɋ a positive regular Borel measure on ܨwith finite moments y୨ = t ୨ dɊ of all orders j אԳ୬ . These yields ଶ

ߣ ߣ ݕା = ߣ ߣ න ݐା ݀ߤ = න ቌ ߣ ݐ ቍ ݀ߤ 0, ,אబ

,אబ

ி

ி

אబ

for any finite subset J ؿԳ୬ and any ൛ɉ୨ ; j אJ ൟ ؿԹ. In other words, any moment sequence ൫y୨ ൯ is positive semidefinite. If ݊ = 1, the converse ୨אԳ

is true since any non-negative polynomial on Թ is a sum of squares. Then, one applies Haviland’s theorem. However, for n 2, there exist positive

30

Chapter One

semi-definite sequences ൫y୨ ൯

୨אԳ

, which are not moment sequences (see

[5]).

4. Markov moment problems and Mazur-Orlicz theorem in concrete spaces The present section is based on the articles [51], [52], [55], the book [58] and the articles [61], [62]. Full and reduced (truncated) moment problems are discussed. A polynomial solution for the multidimensional truncated moment problem is proposed. The first aim of this Section is to point out some cases when the results stated in Section 2 can be applied. A special care is accorded to the control of the norm of the solutions. The first result of this subsection refers to an operator valued Markov moment problem, which is an application of Theorem 2.31 stated above. It refers to a space of continuous functions as the domain space, and it is a multidimensional Markov moment problem over the real field. Let ܪbe a Hilbert space, ܷ a self-adjoint operator acting on ܪ. Let ܻ = ܻ(ܷ) the order complete Banach lattice (which is also a commutative real algebra) of self-adjoint operators discussed in [17, p.303-305]. Namely, if ࣛ = ࣛ( )ܪis the space of all self-adjoint operators acting on ܪ, the natural order relation on ࣛ is, by definition, ܸ ܹ < ܸ݄, ݄ > < ܹ݄, ݄ > ܪ א ݄. With respect to this order relation, ࣛ( )ܪis an ordered Banach space, which is not a lattice. Moreover, the multiplication operation on ࣛ( )ܪis not commutative. Therefore, we use the following notations, to define a suitable subspace ܻ(ܷ) of ࣛ()ܪ: ܻଵ (ܷ) = {ܸ }ܷܸ = ܸܷ ;ࣛ א, ܻ(ܷ) = {ܹ ܻ אଵ (ܷ); ܹܸ = ܸܹ, ܻ א ܸଵ (ܷ)}.

(1.8)

Then ܻ(ܷ) is the codomain space we are interested in. It is an order complete Banach lattice (see [17]) and a commutative real algebra (this last assertion is obvious). Let (ܤ )אԳ be a sequence of operators in ܻ = ܻ(ܷ), and ܤ෨ ܻ אା \{0}. On the other hand, let ݊ ് 0 be a natural number and ܺ be the space of all continuous complex functions ߮ in the unit closed polydisk

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ഥଵ = ൛ݖ( = ݖଵ , … , ݖ ): |ݖ | 1, ݅ { א1, … , ݊}ൟ. ܦ The norm on ܺ is defined by ഥଵ }. ԡ߮ԡஶ = |)ݖ(߮|{ݑݏ: ܦ א ݖ We denote

ഥଵ , |݇| = ݇ଵ + ڮ+ ݇ . ݄ (ݖ = )ݖଵ భ ݖ ڮ , ݇ = (݇ଵ , … , ݇ ) אԳ , ܦ א ݖ Some of the next results are using this space ܻ(ܷ) as the codomain of the involved operators. Let (ܤ )אԳ be a multi - indexed sequence of operators in ܻ = ܻ(ܷ) and ܤ෨ ܻ אା \{0}. Theorem 4.1. (See [51]). Assume that ܣଵ , … , ܣ are elements of ܻ(ܷ) such that there exists a real number > ܯ0, with the property ଶ

|ܤ | ܯ

ଶ

ܣଵ భ ܣ ڮ , א ݇Գ , ܣଶ ܫ, ݇ଵ ! ݇ ! ୀଵ

where ܫ: ܪ՜ ܪis the identity operator. Let {݃ }אԳ ܺ ؿbe such that 1 = ԡ݃ ԡஶ = ݃ (), א ݇Գ . Then there exists a linear bounded operator ܶ א ܺ(ܤ, ܻ) such that ෩ א ݇Գ , ܶ(݄ ) = ܤ , |݇| 1, ܶ(݃ ) ܤ, ܶ(݄) ൫2 + ฮܤ෨ ฮିܯଵ ݁ ିଵ ൯ԡ݄ԡஶ ݑ , ܺ א ݄, ݑ ؔ ܫ݁ܯ. In particular, the following evaluation holds: ԡܶԡ 2 ݁ܯ+ ฮܤ෨ ฮ. Proof. One applies Theorem 2.31, where ܻ = ܻ(ܷ) is defined in (1.8). The subspace generated by {݄ : |݇| 1} stands for ܵ of Theorem 2.31 and the convex hull of the set of the functions ݃ , ݇ אԳ stands for the set ܣ. The following remark is essential ԡ ݏെ ߮ԡஶ |(ݏ0) െ ߮()| = |0 െ 1| = 1, ܵ א ݏ, ܣ א ߮. This proves that ൫ܵ + (ܤ0,1)൯ = ܣ ת, so that (ܤ0,1) stands for ܸ and ԡήԡஶ stands for ܲ of Theorem 2.31. The operator ܤ෨ stands for ݕ. Now let

32

Chapter One

߮ = ߚ ݄ (ܤ ת ܵ א, 1), אబ

where ܬ is a finite subset of Գ . The following relations hold ቮ ߚ ܤ ቮ หߚ หหܤ ห ԡ߮ԡஶ אబ

ݎభ אబ ଵ

אబ

1

ݎ ڮ

หܤ ห,

for any 0 < ݎ < 1, { א 1, … , ݊}, thanks to Cauchy inequalities. Passing to the limit with ݎ ՛ 1, { א 1, … , ݊} and using the fact that ߮ (ܤ א0,1), as well as the hypothesis in the statement, the preceding relation further yields ଶ

ቮ ߚ ܤ ቮ หܤ ห ܯ אబ

אబ

אబ

ଶ

ଶ

ܯቌ భ אԳ

ଶ

ܣଵ భ ܣ ڮ ݆ଵ ! ݆ !

ܣଵ భ ܣ ቍڮቌ ቍ= ݇ଵ ! ݇ ! אԳ

= ݔ݁ܯቌ ܣଶ ቍ ݑ = ܫ݁ܯ = )ܫ(ݔ݁ܯ . ୀଵ

The conclusion is that denoting by ݂: ܵ ՜ ܻ the linear operator which satisfies the moment conditions ݂(݄ ) = ܤ , ݇ אԳ , |݇| 1, we have െ ܫ݁ܯ ݂( )ݏ ݑ = ܫ݁ܯ , (ܤ ת ܵ א ݏ0,1). On the other hand, the following relations hold ܤ෨ ฮܤ෨ ฮ = ܫฮܤ෨ ฮିܯଵ ݁ ିଵ ݑ = ߙଵ ݑ , where ߙଵ = ฮܤ෨ ฮିܯଵ ݁ ିଵ . The conditions on the norms of the functions ݃ , ݇ אԳ lead to ԡ߮ԡ 1, ܣ א ߮.

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So, the constant 1 stands for ߙ from Theorem 2.31. Now all the conditions from the statement of Theorem 2.31 are accomplished. According to the latter theorem, there exists a linear mapping ܶ: ܺ ՜ ܻ, such that ܶ(݄ ) = ݂(݄ ) = ܤ , ݇ אԳ , |݇| 1, ܶ(݃ ) ܤ෨ , א ݇Գ , ܶ(݄) ൫2 + ฮܤ෨ ฮିܯଵ ݁ ିଵ ൯ԡ݄ԡஶ ܫ݁ܯ, ܺ א ݄. From the last inequality, we derive |ܶ(݄)| ൫2 ݁ܯ+ ฮܤ෨ ฮ൯ԡ݄ԡஶ ܫ,

ܺ א ݄.

Since the norm on ܻ is solid, we infer that ԡܶ(݄)ԡ ൫2 ݁ܯ+ ฮܤ෨ ฮ൯ԡ݄ԡஶ ,

֜ ܺ א ݄ԡܶԡ 2 ݁ܯ+ ฮܤ෨ ฮ.

This ends WKHSURRIƑ We continue with another problem that can be solved via Theorem 2.31. We recall that a sequence (ݑ )ஹ in an ordered vector space ܻ is called nonnegative with respect to an interval ॴ كԹ if for any ݊ אԳ, and any coefficients ߣ , ݆ = 0,1, … , ݊, we have ߣ + ߣଵ ݐ+ ڮ+ ߣ ݐ 0 ߣ ฺ ॴ א ݐ ݑ + ߣଵ ݑଵ + ߣ ڮ ݑ ܻ אା . Theorem 4.2. (See [47, Theorem 31, p. 13]). Let [(ܥ0, ܾ]), ߮ ( ݐ = )ݐ , ݆ אԳ, ݆ 1, [ א ݐ0, ܾ],

0 < ܾ אԹ, ܺ =

{߰ }ஹ ܺ ؿ, ԡ߰ ԡ 1, ߰ ؠ, ߰ (0) = 1, ݇ אԳ. Let ܻ be an order complete vector lattice with strong order unit ݑ and (ݕ )ஹଵ a sequence in ܻ such that (ݑ , ݕଵ , ݕଶ , … ) is nonnegative with respect to [0, ܾ]. Then for any ߙଵ ( א0, +λ), there exists a linear operator ܶ ܺ(ܮ א, ܻ) such that ܶ൫߮ ൯ = ݕ , ݆ אԳ, ݆ 1, ܶ(߰ ) ߙଵ ݑ , ݇ אԳ, ܶ(߮) (2 + ߙଵ )ԡ߮ԡݑ , ߮ ܺ א. If we additionally assume that ߙଵ 1, and ܻ is a Banach lattice whose closed unit ball equals the order interval [െݑ , ݑ ], then ԡܶԡ 2 + ߙଵ , ܶ is continuous and positive.

34

Chapter One

Proof. We apply Theorem 2.31 to ܵ = ܵ݊ܽ൛߮ ; ݆ 1ൟ, ܣbeing the convex hull of the set of functions ߰ , ݇ אԳ. Then ԡ െ ܽԡ |(0) െ ܽ(0)| = |0 െ 1| = 1 for all ܵ א and ܽ ܣ א, so that ݀(ܵ, )ܣ 1. This can be written as ൫ܵ + (ܤ, 1)൯ = ܣ ת, so that (ܤ0,1) stands for ܸ and ԡήԡஶ stands for ܲ of Theorem 2.31. On the other hand, ԡ߰ ԡ 1, ݇ אԳ ฺ ԡ߮ԡ 1 for all ߮ ܣ א, so that the number 1 stands for ߙ of Theorem 2.31. We also take in Theorem 2.31 ݕ = ߙଵ ݑ . Define

݂ ቌ ߣ ߮ ቍ = ߣ ݕ , ݊ אԳ, ݊ 1, ߣ אԹ, ݆ = 1, … , ݊. ୀଵ

ୀଵ

Let σୀଵ ߣ ߮ (ܤ ת ܵ א, 1). Using the hypothesis on the sequence which is nonnegative on [0, ܾ], these yields:

ݑݏቐቮ ߣ ݐ ቮ ; [ א ݐ0, ܾ]ቑ 1 ฺ ୀଵ

ߣ ݐ+ 1 0, 1 െ ߣ ݐ 0, [ א ݐ0, ܾ] ฺ ୀଵ

ୀଵ

ߣ ݕ + ݑ , ݑ െ ߣ ݕ ฺ ୀଵ

ୀଵ

݂ ቌ ߣ ߮ ቍ = ߣ ݕ [ אെݑ , ݑ ] ฺ ୀଵ

ୀଵ

݂൫ܵ (ܤ ת, 1)൯ [ كെݑ , ݑ ]. The conclusion is that ݂ satisfies the conditions of Theorem 2.31. According to the latter theorem, there exists a linear extension ܶ of ݂ such that ܶ| ߙଵ ݑ , ܶ(߮) (1 + 1 + ߙଵ )ԡ߮ԡݑ = (2 + ߙଵ )ԡ߮ԡݑ , ߮ ܺ א. Next we prove the last assertion in the statement. To this aim, additionally assume that ߙଵ 1 and ܻ is a Banach lattice. From what is already proved, replacing ߮ by – ߮, we infer that |ܶ(߮)| (2 + ߙଵ )ԡ߮ԡݑ , which further yields: ԡܶ(߮)ԡ (2 + ߙଵ )ԡ߮ԡ, ߮ ܺ א,

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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since the norm on ܻ is solid and ԡݑ ԡ = 1. Hence ܶ is continuous and ԡܶԡ 2 + ߙଵ . Thanks to its continuity and to Weierstrass approximation theorem, to prove the positivity of ܶ, it is sufficient to show that ܶ( ) for all polynomials , with )ݐ( 0 for all [ א ݐ0, ܾ]. Indeed, let = )ݐ( ߣ + σୀଵ ߣ ݐ 0, [ א ݐ0, ܾ]. According to our hypothesis, this implies

ߣ ݑ + ߣ ݕ ฺ ߣ ݕ െߣ ݑ ฺ ୀଵ

ୀଵ

ܶ ቌߣ + ߣ ߮ ቍ = ߣ ܶ(߰ ) + ߣ ݕ ୀଵ

ୀଵ

ߣ ߙଵ ݑ െ ߣ ݑ = ߣ (ߙଵ െ 1)ݑ . The proof is complete. Corollary 4.3. (See [47, Corollary 32, p. 14]). Let ܪbe a Hilbert space, ܷ )ܪ(ࣛ אa positive self-adjoint operator acting on ܪ, and ܻ = ܻ(ܷ) the order complete Banach lattice defined by (1.8). Assume that the spectrum ߪ(ܷ) = [0, ܾ] for some positive real number ܾ and let ߙଵ be such that ߙଵ 1. There exists a nondecreasing mapping ߪ: [0, ܾ] ՜ ܻ such that

න ݐ ݀ߪ = ܷ , ݆ אԳ, ݆ 1, න ݁ ି௧ ݀ߪ ߙଵ ܫ, ݇ אԳ, න ߮(ߪ݀ )ݐ

(2 + ߙଵ )ԡ߮ԡܫ, ߮ [(ܥ א0, ܾ]). For such a mapping ߪ, the following inequality holds: ߙଵ + 1 ԡߪ(ܾ) െ ߪ(0)ԡ + ԡ݁(ݔെܷ݇)ԡ, ݇ אԳ. Proof. We apply Theorem 4.2 to ܻ = ܻ(ܷ), ݑ = ܻ א ܫ, ݕ = ܷ , ݆ { א1,2, … }, ߰ ((ݔ݁ = )ݐെ݇)ݐ, ݇ אԳ, [ א ݐ0, ܾ], ߮ ( ݐ = )ݐ , ݆ אԳ, ݆ 1, [ א ݐ0, ܾ]. Here ܫ: ܪ՜ ܪis the identity operator. The sequence (ܫ, ܷ, ܷ ଶ , … , ܷ , … ) is nonnegative with respect to the interval [0, ܾ]. Indeed, if ߣ = )ݐ( + ߣଵ ݐ+ ڮ+ ߣ ݐ 0 [ א ݐ0, ܾ], then, this inequality holds for all א ݐ ߪ(ܷ) = [0, ܾ]. Consequently, we obtain:

36

Chapter One

ߣ + ߣଵ ܷ + ڮ+ ߣ ܷ = න ܧ݀)ݐ( , ఙ()

since the spectral measure ݀ܧ is positive. Thus, all the requirements in the statement of Theorem 4.2 are satisfied, so that there exists a linear positive (bounded) operator ܶ: [(ܥ0, ܾ]) ՜ ܻ, with the properties specified in the conclusion Theorem 4.2. According to [17, p. 272], there is a nondecreasing mapping ߪ: [0, ܾ] ՜ ܻ such that

ܶ(߮) = න ߮(ߪ݀)ݐ, ߮ [(ܥ א0, ܾ]).

This leads to:

න ݐ ݀ߪ = ܶ൫߮ ൯ = ܷ , ݆ אԳ, ݆ 1, න ݁(ݔെ݇߰(ܶ = ߪ݀ )ݐ ) ߙଵ ܫ,

݇ אԳ, න ߮( )߮(ܶ = ߪ݀ )ݐ (2 + ߙଵ )ԡ߮ԡܫ,

for all ߮ [(ܥ א0, ܾ]). To prove the last inequality in the statement, from the previously mentioned inequality, for any ݇ אԳ we derive:

ஶ

ߙଵ ܫ න ݁ ି௧ ݀ߪ = න ൭1 + ஶ

ߪ(ܾ) െ ߪ(0) + ୀଵ

ୀଵ

(െ݇) ݐ൱ ݀ߪ = ݉!

(െ݇) ܷ = ߪ(ܾ) െ ߪ(0) െ ܫ+ ݁(ݔെܷ݇) ฺ ݉!

(ߙଵ + 1) ܫ ߪ(ܾ) െ ߪ(0) + ݁(ݔെܷ݇). Because the norm on the Banach lattice ܻ is monotone (increasing) on the positive cone ܻା , the preceding inequality implies ߙଵ + 1 ԡߪ(ܾ) െ ߪ(0)ԡ + ԡ݁(ݔെܷ݇)ԡ, ݇ אԳ. 7KHSURRILVFRPSOHWHƑ

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The next results refer to the Markov moment problem on the space )ܭ(ܥ, where ܭis an arbitrary compact Hausdorff topological space. These theorems are based on statements proved in [51], [52], [55], [58]. Theorem 4.4. Let ܭbe a compact Hausdorff topological space, ߤ a positive regular Borel measure defined on the class of Borel subsets of ܭ, )ܭ(ܥthe Banach lattice of all real valued continuous functions on ܭ, ൛߮ ൟא a family of linearly independent elements in )ܭ(ܥ, ൛ݕ ൟא a given family of elements in ܮஶ ఓ ()ܭ. The following two statements are equivalent (a) there exists a linear (positive) bounded operator ܶ: )ܭ(ܥ՜ ܮஶ ఓ ( )ܭsuch that ܶ൫߮ ൯ = ݕ , ݆ ܬ א, ܶ(߮) ൬)ݐ(߮ ݑݏ൰ ¶ )ܭ(ܥ א ߮. ௧א

(1.9)

In particular, the following equalities hold ܶ(¶) = ¶, ԡܶԡ = 1; (b) for any finite subset ܬ ܬ ؿand any ൛ߙ ; ݆ ܬ א ൟ ؿԹ, the following relation holds true ߙ ݕ ݑݏቌ ߙ ߮ ()ݐቍ ¶ אబ

௧א

(1.10)

אబ

Proof. The implication (ܽ)֜ (b) is obvious, thanks to the properties of ܶ. To prove (ܾ) ֜ (ܽ), one applies Theorem 2.29, implication (ܿ) ֜ (ܽ), for ܺ = )ܭ(ܥ, ܻ = ܮஶ ఓ ()ܭ, ܲ(߰) = ൬)ݐ(߰ ݑݏ൰ ¶, ߰ ܺ א. ௧א

(1.11)

Observe that ܲ defined by (1.11) is a scalar valued sublinear nondecreasing functional multiplied by the class of the constant function ¶ in ܮஶ ఓ ()ܭ, hence is an isotone (i.e., monotone increasing) sublinear operator. The inequality (1.10) is equivalent to the fact that condition written at point (c) of Theorem 2.29 is accomplished. Since ܮஶ ఓ ( )ܭis an order complete vector lattice, according to Theorem 2.29, there exists a positive linear operator ܶ: ܺ ՜

38

Chapter One

ܮஶ ఓ ( )ܭwith the properties mentioned at point (a) of the latter theorem. It results ܶ൫߮ ൯ = ܶ ൫߮ ൯ ؔ ݕ , ݆ ܬ א. Moreover the following implications hold ߮ )߮(ܶ ֜ ܧ א ൬)ݐ(߮ ݑݏ൰ ¶, െܶ(߮) ൬ ݑݏെ߮()ݐ൰ ¶ = ௧א

௧א

= െ ൬݂݅݊ ߮()ݐ൰ ¶ ֜ ൬݂݅݊ ߮()ݐ൰ ¶ ܶ(߮) ൬)ݐ(߮ ݑݏ൰ ¶ ௧א

௧א

௧א

Thus (1.9) are proved. In particular, for ߮ = ¶, it results ܶ(¶) = ¶. Since any ߮ ܺ אwith ԡ߮ԡ 1 is situated in the order interval [െ¶, ¶], the positivity of ܶ leads to ܶ(߮) (ܶ[ אെ¶), ܶ(¶)] = [െ¶, ¶] ֜ ԡܶ(߮)ԡ 1 ֜ ԡܶԡ 1. But we have already seen that ԡܶ(¶)ԡ = ԡ¶ԡ = 1. Hence ԡܶԡ = 1 and the proof is completeƑ In the next theorem, ܭwill be a compact subset of Թ (݊ 1 is a natural number), ܺ = )ܭ(ܥ, ܻ an order complete Banach lattice with a strong order unit ¶ such that the order interval [െ ¶ , ¶ ] is equal to the closed unit ball of ܻ, ݆ = (݆ଵ , … , ݆ ) אԳ . ݐ( = ݐଵ , … , ݐ ) ܭ א, |݆| = σୀଵ ݆ , ߮ (= )ݐ ݐ = ݐଵభ ݐ ڮ . Theorem 4.5. Let ൛ݕ ; |݆| ݉ൟ ܻ ؿ, ݉ אԳ, ݉ 1. The following statements are equivalent: (a) there exists a positive linear operator ܶ: )ܭ(ܥ՜ ܻ such that ܶ൫߮ ൯ = ݕ , |݆| ݉, ܶ(߮) ൬)ݐ(߮ ݑݏ൰ ¶ ௧א

)ܭ(ܥ א ߮, ԡܶԡ = 1;

(1.12)

(b) for any ൛ߚ ; |݆| ݉ൟ ؿԹ, the following relation holds

ߚ ݕ ቌ ݑݏቌ ߚ ݐ ቍቍ ¶ אԳ ||ஸ

௧א

(1.13)

||ஸ

Proof. One repeats the proof of Theorem 4.4, where we replace ܮஶ ఓ ( )ܭby

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ܻ, ߮ ( ݐ = )ݐ , ܭ א ݐ, ݆ = ܬ אԳ , |݆| ݉, ܲ(߰) = ൬)ݐ(߰ ݑݏ൰ ¶ , ௧א

߰ )ܭ(ܥ = ܺ א. Some of the notations have been defined before the statement. Clearly, from (1.12) with positive and unital ܶ, the relation (1.13) follows. For the converse, repeating the arguments from the proof of Theorem 4.4, the existence of a positive linear operator verifying (1.12) follows (via Theorem 2.29, (c)֜(a)). Here the subspace ܵ = ܯ൛߮ ; ݆ א Գ , |݆| ݉ൟ is the vector subspace of all polynomial functions on ܭ, of degree ݉. From (1.12), in particular, ܶ(¶ ) = ¶ follows as well, and the proof is completeƑ When applying Mazur-Orlicz theorem (Theorem 2.33), one can work with the subspace of all polynomial functions on ؿ ܭԹ , without any restriction on their degree (one proves a full Mazur-Orlicz theorem). Such a result is not a direct consequence of the density of polynomials in )ܭ(ܥ (unlike the case of the full moment problem for )ܭ(ܥ, ؿ ܭԹ ). In Theorem 4.5, a solution for a truncated moment problem was proposed. A linear operator ܶ from )ܭ(ܥto ܻ is called a Markov operator if ܶ is positive and ܶ(¶) = ¶ (the definition is valid for any Hausdorff compact topological space )ܭ. It is easy to observe that a linear operator ܶ )ܭ(ܥ(ܮ א, ܻ) is a Markov operator if and only if ܶ(߮) ൬)ݐ(߮ ݑݏ൰ ¶ )ܭ(ܥ א ߮. In ௧א

particular, solutions ܶ from Theorems 4.4, 4.5 and 4.6 (the latter being proved below) are Markov operators. Let ܻ be an order complete Banach space, having a strong order unit ¶ , ൫ݕ ൯אԳ a sequence in ܻ. We prove the following theorem. Theorem 4.6. Let ܭ = ܭଵ × ܭ × ڮ ؿԹା be such that ܭ ؿԹା is compact and denote

ݎ = ܭݑݏ , ݈ = 1, … , ݊, ݎ = ݎଵ భ ݎ ڮ , ݆ = (݆ଵ , … , ݆ ) אԳ . The following statements are equivalent: (a) there exists a (positive) linear operator ܶ: )ܭ(ܥ՜ ܻ such that ܶ൫߮ ൯ ݕ , ݆ אԳ , ܶ(߮) ൬)ݐ(߮ ݑݏ൰ ¶ ௧א

)ܭ(ܥ א ߮, ԡܶԡ = 1;

40

Chapter One

(ܾ) ݕ ݎ ¶ א ݆Գ . Proof. The implication (a)ฺ(b) is obvious, thanks to the properties of ܶ. Namely, the following relations hold true ݕ ܶ൫߮ ൯ ൬߮ ݑݏ ()ݐ൰ ¶ = ௧א

൬ݐ(ݑݏଵభ ݐ ڮ )൰ ¶ = ݎ ¶ , ௧א

݆ אԳ .

To prove (b)֜(a), we use the implication (c)֜(a) of Theorem 2.33. The conditions mentioned at (c) of the latter theorem is accomplished, since for any finite subset ܬ ؿԳ the following inequalities hold ݕ ݎ ¶ , ߣ 0, ܬ א ݆ ֜ ߣ ݕ ቌ ߣ ݎ ቍ ¶ = ቌ( ߣ ݐ )|௧ୀ(భ,…, ) ቍ ¶ אబ

אబ

אబ

= ݑݏቌ ߣ ݐ ቍ ¶ = ܲ ቌ ߣ ߮ ቍ, ௧א

אబ

ܲ(߰) ؔ ቆ)ݐ(߰ݑݏቇ ¶ ,

אబ

߰ )ܭ(ܥ א.

௧א

According to Theorem 2.33, (c)֜(a), there exists a positive linear operator ܶ: )ܭ(ܥ՜ ܻ with the properties mentioned at point (a) of the present theorHP7KHSURRILVFRPSOHWHƑ Theorem 4.7. Let ܺ be a Banach lattice, ܻ an order complete Banach lattice, ൛߮ ൟא ܺ ؿା , ൛ݕ ൟא ܻ ؿ, ܩa linear positive (bounded) operator from ܺ into ܻ, and ߙ a positive number. The following statements are equivalent: (a) there exists a linear positive bounded operator ܶ ܤ אା (ܺ, ܻ), such that ܶ൫߮ ൯ ݕ , ܬ א ݆, ܶ( )ݔ ߙ)|ݔ|(ܩ, ܺ א ݔ, ԡܶԡ ߙԡܩԡ; (b) ݕ ߙܩ൫߮ ൯, ܬ א ݆.

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Proof. (a)֜(b) is obvious, because of ݕ ܨ൫߮ ൯ ߙܩ൫ห߮ ห൯ = ߙܩ൫߮ ൯, ܬ א ݆. For the converse, we apply Theorem 2.33, (b)֜(a). Let ܬ ܬ ؿbe a finite subset, ൛ߣ ൟא ؿԹା , ܺ א ݔ, such that σאబ ߣ ߮ ݔ. బ

Then using (b) and the fact that the scalars ߣ are nonnegative, as well as the positivity of ܩ, we derive

ߣ ݕ ߙ ߣ ܩ൫߮ ൯ = ߙ ܩቌ ߣ ߮ ቍ אబ

אబ

אబ

ߙ )ݔ(ܩ ߙ)ݔ(ܲ ؔ )|ݔ|(ܩ. Application of Theorem 2.33 leads to the existence of a linear positive operator ܶ from ܺ into ܻ such that ܶ൫߮ ൯ ݕ , ܬ א ݆, ܶ( )ݔ ߙ)|ݔ|(ܩ, ܺ א ݔ. From the last relation, also using the fact that the norms on ܺ and ܻ are solid (| |ݑ | ֜ |ݒԡݑԡ ԡݒԡ), we deduce |ܶ( |)ݔ ߙ ֜ )|ݔ|(ܩԡܶ()ݔԡ ߙԡܩԡԡ||ݔԡ = ߙԡܩԡԡݔԡ, ܺ א ݔ. It follows that ԡܶԡ ߙԡܩԡ. This concludes the proof. Ƒ Corollary 4.8. Let ܯbe a measure space, ߤ a positive measure on ܯ, ߤ( < )ܯλ, ܺ = ܮఓ ()ܯ, 1 < λ, ݃ 0 an element of ܮఓ ()ܯ, where ( א ݍ1, λ] is the conjugate of ( 1Τ + 1Τ = ݍ1), ߙ a positive number. Let ൛߮ ൟא , ൛ݕ ൟא be as in Theorem 4.7, where ܻ = Թ. The following statements are equivalent:

(a) there exists ݄ ܮ אఓ ()ܯ, 0 ݄ ߙ݃ a.e., ெ ݄߮ ݀ߤ ݕ , ;ܬ א ݆ (b) ݕ ߙ ெ ݃߮ ݀ߤ, ܬ א ݆. Proof. One applies Theorem 4.7 for = )߰(ܩெ ݃߰ ݀ߤ, ߰ ܺ א, ܻ = Թ, as well as the representation of linear positive continuous functionals on ܮ spaces by means of nonnegative elements from ܮ spaces. To prove (b)֜(a),

42

Chapter One

from the preceding results it follows that there exists ݄ אቀܮఓ ()ܯቁ such ା

that ெ ݄߮ ݀ߤ ݕ , ܬ א ݆and

න ݄߰݀ߤ ߙ න ݃߰݀ߤ ெ

ெ

for all nonnegative functions ߰ ܮ אఓ ()ܯ. Now we choose ߰ = ߯ , where ܤis an arbitrary measurable subset of ܯ. Then the last relation can be rewritten as න (݄ െ ߙ݃) ݀ߤ 0

for all such subsets ܤ. A straightforward application of a measure theory argument [68] leads to ݄ െ ߙ݃ 0 a.e. in ܯ. Since (a)֜(b) is obvious, this concludes the proof. ᇝ Corollary 4.9. Let consider the measure space = ܯԹା , ݊ { א1,2 … }, endowed with the measure ݀ߤ = ݁ݔ൫െ σୀଵ ݐ ൯݀ݐଵ ݐ݀ ڮ , > 0, א ݆ {1, … , ݊}, ߙ a positive number. The following statements are equivalent: (a) there

exists

݄ ܮ אஶ ఓ (Թା ), Թ ݄ ߤ݀ ݐ ݕ , א ݆Գ , 0 ݄ ߙ శ

almost everywhere. (b) The following inequalities hold: ݕ ߙ

భ !ڮ! ೕ శభ ೕ శభ ڮ

భభ

, ݆( = ݆ଵ , … , ݆ ) אԳ .

Proof. One applies Corollary 4.6 to = 1, = ݍλ, ݃ = a.e. The notation ݐ is the multi – index notation ݐ = ݐଵభ ݐ ڮ . The conclusion follows via Fubini theorem and Gamma function’s SURSHUWLHVƑ

Theorem 4.10. Let ܺ = ܮఓ ()ܯ, 1 < < λ, ߤ 0, ߤ( < )ܯλ, ൛߮ ൟא ؿ ܺ, ൛ݕ ൟא ؿԹ, ߙ > 0, ߙ אԹ, ݍthe conjugate of . Consider the following statements:

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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(a) there exists ݄ אቀܮఓ ()ܯቁ such that ା

න ݄߮ ݀ߤ ݕ , ܬ א ݆, න ݄߰݀ߤ ߙԡ߰ԡ ൫ߤ()ܯ൯ ெ

ଵΤ

;ܺ א ߰

,

ெ

(b) we have ݕ ߙ ெ ߮ ݀ߤ, ܬ א ݆. Then (b)֜(a) holds. Proof. Let ܬ ܬ ؿbe a finite subset, ൛ߣ ൟא ؿԹା . Hölder inequality and బ

using also (b), lead to the following implications

ߣ ߮ ߰ ֜ න ቌ ߣ ߮ ቍ ݀ߤ න ߰݀ߤ ԡ߰ԡ ൫ߤ()ܯ൯ אబ

אబ

ெ

֜

ெ ଵΤ

ߣ ݕ ߙ න ቌ ߣ ߮ ቍ ݀ߤ ߙ ԡ߰ԡ ൫ߤ()ܯ൯ אబ

ଵΤ

= ܲ(߰).

אబ

ெ

Application of Theorem 2.33 and measure theory arguments yield the existence of ݄ ܮ אఓ ( )ܯsuch that

ܶ൫߮ ൯ = න ݄߮ ݀ߤ ݕ , ܬ א ݆, ெ ଵΤ

ܶ(߰) = න ݄߰݀ߤ ߙԡ߰ԡ ൫ߤ()ܯ൯

, ߰ ܺ א.

ெ

Moreover, since )߰(ܨ 0, ܺ א ߰ା , we have

න ݄߰݀ߤ 0, ܺ א ߰ା . ெ

Taking ߰ = ߯ , where ܯ ؿ ܤis a measurable set such that ߤ( > )ܤ0, one obtains

44

Chapter One

න ݄݀ߤ 0

for all such subsets ܤ. Application of a well-known measure theory argument leads to ݄ 0 ߤ െ ܽ. ݁. From the previous relations we also derive that ԡ݄ԡ ߙ൫ߤ()ܯ൯

ଵΤ

. ThiVFRQFOXGHVWKHSURRIƑ

The following theorem represents an application of the general result stated in Theorem 2.33 to some other concrete spaces ܺ, ܻ. Let ܪbe an arbitrary Hilbert space, ܷ a positive self - adjoint operator acting on ܪ, ൫ܤ ൯אԳ a sequence in ܻ, where ܻ = ܻ(ܷ) is defined by (1.8). We denote by ܧ be the spectral measure attached to ܷ and by ߪ(ܷ) the spectrum attached to ܷ. Let denote by ߮ , ݆ אԳ the basic polynomials ߮ ( ݐ = )ݐ , ݆ אԳ, א ݐ ߪ(ܷ), ܺ ؔ ܥ൫ߪ(ܷ)൯. Theorem 4.11. The following statements are equivalent: (a) there exists a linear bounded positive operator ܶ ܤ אା (ܺ, ܻ) such that ܶ൫߮ ൯ ܤ , ݆ אԳ, ܶ(߮) න |߮| ݀ܧ , ܺ א ߮, ԡܶԡ 1; ఙ()

(b) ܤ ܷ , ݆ אԳ. Proof. The implication (a)֜(b) is obvious:

ܤ ܶ൫߮ ൯ න ห߮ ห݀ܧ = න ߮ ݀ܧ = ܷ , ݆ אԳ; ఙ()

ఙ()

(we have used the positivity of the operator ܣwhich leads to ห߮ ห = ߮ on ߪ(ܷ) ؿԹା ). For the converse, one applies Theorem 2.33 (b)֜(a), where ܬ stands for Գ, ݔ stands for ߮ and ݕ stands for ܤ , א ݆Գ. Let ܬ and ൛ߣ ൟא be as mentioned at point (b) of Theorem 2.33. The following బ

implications hold

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ߣ ߮ ߮ ֜ ܺ א ߣ න ߮ ݀ܧ = ߣ ܷ න ߮݀ܧ אబ

אబ

אబ

ఙ()

ఙ()

න |߮| ݀ܧ ؔ ܲ(߮). ఙ()

The positivity of the spectral measure ݀ܧ has been used. On the other hand, the hypothesis (b), the fact that the scalars ߣ are nonnegative and the preceding evaluation, yield ߣ ܤ ߣ ܷ ֜ ݆ ߣ ܤ ߣ ܷ ܲ(߮), אబ

אబ

where ܲ(߮) was defined above. Thus, the implication (b) of Theorem 2.33 is satisfied. Application of the latter theorem leads to the existence of a “feasible solution” ܶ having the property mentioned at point (a) of the present theorem. The last property is a consequence of the preceding one, using the fact that the norm on ܻ is solid. Namely,

±ܶ(߮) = ܶ(±߮) න |߮| ݀ܧ ؔ ܲ(߮) ฺ ఙ()

|ܶ(߮)| ܲ(߮) ฺ ԡܶ(߮)ԡ ԡܲ(߮)ԡ ԡ߮ԡ ή ԡܫԡ = ԡ߮ԡ, ߮ ܺ א. Hence ԡܶԡ 1. 7KLVFRQFOXGHVWKHSURRIƑ Remark 4.1. If in Theorem 4.11 one additionally assumes that ԡܷԡ < 1, then for any self - adjoint operators satisfying (b) one has ܤ ܷ = ( ܫെ ܷ)ିଵ . אԳ

אԳ

The next results of this section refer to the truncated moment problem. The way of writing is for the multidimensional case, although the onedimensional variant is important. Assume now that the moments ݉ , where ݇ = (݇ଵ , … , ݇ ) אԳ , 0 ݇ ݀, ݅ { א1,2, … , ݊} are exact, determined in a training stage, when no external influence can occur. Also assume that the moments in real time stage, denoted by ݕ , where ݇

46

Chapter One

݀, ݅ = 1, … , ݊, can be measured, but errors may occur due to external influences. The vector subspace of polynomials involved this way has the dimension (݀ + 1) . Assume now that there exists ݄, ݓin ܮଶఓ ( )ܨsuch that න ݐ ݄(݉ = ߤ݀)ݐ , න ݐ ݕ = ߤ݀)ݐ(ݓ ி

ி

for all multi-index ݇ אԳ , having all components 0 ݇ ݀, ݅ = 1, … , ݊. Let {݁ }ஸஸௗ be the orthogonal system of polynomials having unit norms, obtained from the system {߮ }ஸ ஸௗ , ߮ ( ݐ = )ݐ , via the Gram-Schmidt process, in the space ܮଶఓ ()ܨ. We denote by ܲௗ : ܮଶఓ ( )ܨ՜ ܵ݁{݊ܽ ; ݇ א {0, … , ݀}, ݅ = 1, … , ݊}, the orthogonal projection. The following equalities hold: ݁ = ܿ, ߮ , ஸ

where ݈ ݇ means ݈ ݇ , ݅ = 1, … , ݊, and the coefficients ܿ, are known from the Gram-Schmidt process. This yield: < ܲௗ (݄) െ ܲௗ ()ݓ, ݁ > = ܿ, < ܲௗ (݄) െ ܲௗ ()ݓ, ߮ > ஸ

= ܿ, (݉ െ ݕ ). ஸ

Hence ԡܲௗ ( )ݓെ ܲௗ (݄)ԡଶଶ,ఓ = < ܲௗ (݄) െ ܲௗ ()ݓ, ݁ >ଶ = ஸௗ

ଶ

ቌ ܿ, (݉ െ ݕ )ቍ ஸௗ

ஸ

ଶ ൱ ή ൭(݉ െ ݕ )ଶ ൱. ൭ ܿ, ஸௗ

ஸ

ஸௗ

ஸ

We have proved the following result:

(1.14)

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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Proposition 4.12. The integral mean of the square ൫ܲௗ ( )ݓെ ܲௗ (݄)൯ in the first line of the above relations (1.14), can be evaluated in terms of the squares of the errors |݉ െ ݒ |, (݇ ݀ for all ݅ { א1, … , ݊}), as the last sums in (1.14) show. In the end of this section, we propose a polynomial solution for the truncated multidimensional scalar-valued moment problem [62], completing a result of [61]. Related evaluation for the norm of the vector formed with the coefficients of the polynomial solution is outlined. The general idea is to replace the space ࣪ of all polynomial functions on a closed subset ك ܨԹ , ݊ { א2,3, … } having nonempty interior, with the subspace generated by the monomials

߮ ( ݐ = )ݐ = ݐଵభ ݐ ڮ , ݐ( = ݐଵ , … , ݐ ) ܨ א, ݆ = (݆ଵ , … , ݆ ), ݆ { א0,1, … , ݀}, ݇ = 1, … , ݊, where ݀ 1 is a fixed integer. Let ݓbe a continuous positive real valued function on ܵ such that all the absolute moments න| |ݐ ݐ݀ )ݐ(ݓ, ݆ = (݆ଵ , … , ݆ , … , ݆ ) אԳ , 0 ݆ ݀, ݇ = 1, … , ݊, ௌ

are finite (here | |ݐ = |ݐଵ | భ ݐ| ڮ | , ݀ݐ݀ = ݐଵ ݐ݀ ڮ ). Being given a finite set of numbers ൛ݕ ; ݆ אԳ , ݆ ݀, ݇ = 1, … , ݊ൟ, we are looking for a solution ݄ of the moment interpolation problem න ݐ ݄(ݕ = ݐ݀ )ݐ(ݓ)ݐ , 0 ݆ ݀, ݇ = 1, … , ݊.

(1.15)

ௌ

The simplest function ݄ satisfying equalities (1.15) is a polynomial ݄( = )ݐ ߣ ݐ , ܵ א ݐ,

(1.16)

ஸೖ ஸௗ, ୀଵ,…,

where ݈ = (݈ଵ , … , ݈ , … , ݈ ) אԳ . The number of terms of the sum in (1.16) is at most ܰ = (݀ + 1) , and ܰ is the dimension of the subspace of polynomials involved in this problem. We must determine the unknown

48

Chapter One

coefficients ߣ , 0 ݈ ݀, ݇ = 1, … , ݊, such that ݄ be a solution of (1.15). The corresponding result gives the explicit form of the solution and related evaluations of its norm. These inequalities do not involve computing the inverse of the matrix ( ܣsee the proof of the next theorem). Theorem 4.13 is formulated as follows: Theorem 4.13. The vector ࣅ = ൫ߣ ൯ஸೖ ஸௗ, of the unknowns ߣ , 0 ݈ ଵஸஸ

݀, ݇ = 1, … , ݊ defined by (1.15) and (1.16) is given by (1.19), where the matrix ܣis defined by (1.18). This matrix is positive definite and evaluations (1.20), (1.21) hold. Proof. Inserting ݄( )ݐdefined by (1.16) in (1.15), it is easy to see that the necessary and sufficient conditions required on ߣ , ݈ ݀, ݇ = 1, … , ݊ are: ߣ න ݐା ݕ = ݐ݀)ݐ(ݓ , ݆ ݀, ݇ = 1, … , ݊. ஸೖ ஸௗ, ୀଵ,…,

(1.17)

ௌ

This is a linear system with the unknowns ߣ , 0 ݈ ݀, ݇ = 1, … , ݊, ݈ = (݈ଵ , … , ݈ ), consisted of ܰ = (݀ + 1) equations with ܰ unknowns. The matrix of the system is the ܰ × ܰ symmetric matrix

= ܣ൫ܽ, ൯ஸ

ೖ ,ೖ ஸௗ

, ܽ, = න ݐା ݐ݀)ݐ(ݓ, ௌ

0 ݆ ݀, 0 ݈ ݀, ݇ = 1, … , ݊.

(1.18)

The main property of the matrix ܣis that is positive definite. Indeed, according to (1.18), the following relations hold true: ଶ

ܽ, ߣ ߣ = න ቌ ߣ ݐቍ > ݐ݀)ݐ(ݓ0 ஸೖ ஸௗ, ஸೖ ஸௗ

ௌ

ஸೖ ஸௗ

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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for all not null vectors ࣅ = ൫ߣ ൯ஸೖ ஸௗ, . As a consequence, all eigenvalues ଵஸஸ

of the matrix ܣare positive. In particular, 0 is not in the spectrum of ܣ, so that this matrix is invertible, and its inverse is also positive definite. Since (1.17) can be written as ࣅܣ௧ = ௧ , we infer the unique solution ࣅ௧ = ିܣଵ ௧ .

(1.19)

Hence the problem is reduced to the computation of ିܣଵ . Related evaluation might be useful. For example, if we denote by ȍ the greatest eigenvalue and by ߱ the smallest eigenvalue of ܣ, then ԡࣅԡଶ ԡିܣଵ ԡԡԡଶ = (1Τ߱ )ԡԡଶ ,

(1.20)

where ԡ ԡ is the Euclidean norm on Թே . Similarly, from the inequality ିܣଵ ( غ1Τȍ )ܫ, also using (1.19) and Cauchy-Schwarz inequality, the following evaluations hold: ԡࣅԡଶ ԡԡଶ < ࣅ, > = < ିܣଵ , > (1Τȍ )ԡԡଶଶ . It results ԡࣅԡଶ (1Τȍ )ԡԡଶ

(1.21)

7KHSURRILVFRPSOHWHƑ For other results on construction of a solution of some truncated moment problems see [23], [24], [41] and Section 4 of [47]. Next we discuss some of the results proved in Section 4 of [47] (see also the references therein). Let ܾ ( א0, λ). We denote ߮ ( ݐ݆ = )ݐିଵ , [ א ݐ0, ܾ], ݆ = 1, … , ݊. The ܮଵ ([0, ܾ]), ܮஶ ([0, ܾ]) spaces are considered with respect to Lebesgue measure on [0, ܾ].

50

Chapter One

Theorem 4.14. For a given family of numbers ൫݉ ൯ୀଵ , consider the following statements (a), (b) and (c): (a) There exists ݄ ܮ אஶ ([0, ܾ]), 0 ݄( )ݐ 1 ିଵ [0, ܾ], ݆ݐ ݄(݉ = ݐ݀)ݐ , ݆ = 1, … , ݊.

a.

e.

in

(b) For any family of scalars ൫ߣ ൯ୀଵ , we have: σୀଵ ߣ ݉ σୀଵ ߣ ܾ . (c) There exists a Borel subset [ ك ܤ0, ܾ] such that ݆ ݐିଵ ݀= ݐ ݉ , ݆ = 1, … , ݊. The implications (ܾ) ฺ (ܽ) (ܿ) hold. Proof. To prove (b) ฺ (a), we apply Theorem 2.32. Namely, if

ୀଵ

ߣ ݆ ݐିଵ = (߰ଶ െ ߰ଵ )(ܽ )ݐ. ݁. , ߰ଵ , ߰ଶ ܮ( אଵ ([0, ܾ]))ା ,

then integrating on [0, b] yields

ୀଵ

ߣ ܾ න ߰ଶ (= ݐ݀ )ݐ: ܶଶ (߰ଶ ) = ܶଶ (߰ଶ ) െ ܶଵ (߰ଵ ), ܶଵ ؠ .

According to theorem 2.32, there exists a linear functional ܶ: ܮଵ ([0, ܾ]) ՜ Թ, such that

ܶ൫߮ ൯ = ݉ , ݆ = 1, … , ݊, 0 ܶ(߰) න ߰( ݐ݀)ݐ, ߰ ܮ( אଵ ([0, ܾ]))ା .

For an arbitrary ߮ ܮ אଵ ([0, ܾ]), we have

|ܶ(߮)| ܶ(߮ ା ) + ܶ(߮ ି ) = ܶ(|߮|) න |߮( = ݐ݀|)ݐԡ߮ԡଵ .

Measure theory arguments lead to the existence of a function ݄ א ܮஶ ([0, ܾ]), 0 ݄ 1, ܶ(߮) = ݄߮݀ ݐ, ߮ ܮ אଵ ([0, ܾ]). The proof of (ܾ) ฺ (ܽ) is complete. The implication (ܿ) ฺ (ܽ) is obvious: if (c) holds, we can take as ݄ appearing at point (a) as the characteristic function ߯ of the Borel subset ܤ. In order to prove (ܽ) ฺ (ܿ) we use the equality (15.14) of [26], p. 109, which in turn is based on the Lyapunov convexity theorem

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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(see the Corollary on page 108 of [26]; see also [26, Exercise 2.57, p. 117]). The control vector )ݐ(ݑ = ݑappearing in [26] can be taken as the ݊ dimensional vector ൫݄()ݐ, … , ݄()ݐ൯ and the matrix ߖ = ߖ( )ݐwill be the ݊ × ݊ diagonal matrix defined by ߖ, ( ݐ݆ = )ݐିଵ , ݅ = ݆, ߖ, ( = )ݐ0, ݅ ് ݆,

݅, ݆ = 1, … , ݊.

Finally, observe that if )ݐ(ݓ = ݓis in the closed unit ball of the space ܮஶ ([0, ܾ]) and takes values in {െ1,1} (because is an extremal element of ଵ the unit ball), then = )ݐ(ݒ൫1 + )ݐ(ݓ൯ is in the intersection of the closed ଶ unit ball with the positive cone of ܮஶ ([0, ܾ]), and takes values in {0,1}, and hence is the characteristic function of the set {[ א ݐ0, ܾ]; = )ݐ(ݓ1}. The conclusion follows and the proof is complete. ᇝ Corollary 4.15. Under the equivalent assertions (a), (c) of Theorem 4.14, there are sequences ݕଵ, < ݔଵ, < ݕଶ, < ݔଶ, < ݕ < ڮ, < ݔ, < ڮ, ݊ אԳ, such that the following relations hold: ஶ െ ݕ, ൯ቍ , ݇ = 1, … , ݊. ݉ = ݂݅݊ ቌ൫ݔ, אԳ

ୀଵ

Proof. One uses the fact that any Borel subset is the union of a ܩఋ set and DVHWRIQXOO/HEHVJXHPHDVXUHƑ Now we go on with the solution of the full moment problem regarded as the weak limit of solutions of the corresponding sequence of truncated moment problems. To this aim, we partially use an idea of J. Stochel (see reference [26] in the list of references of our paper numbered as [47] in the Bibliography of the present book). Theorem 4.16. (See [47, Theorem 28, p. 11-12]). With the notations of Theorem 4.14, let 0 ܽ < ܾ < λ and let (݉ )ஹଵ be a sequence of real numbers. Consider the following statements:

52

Chapter One

(a) There exists a Borel function ݄ such that

0 ݄( )ݐ 1 ܽ. ݁. , ݉ = න ݆ ݐିଵ ݄(ݐ݀)ݐ, ݆ אԳ, ݆ 1.

(b) For any natural number ݊ 1 and any ߝ > 0, there exist nonnegative scalars ߚ , ݆ = 1, … , ݊, and sequences: ݕ,ଵ < ݔ,ଵ < ݕ < ڮ, < ݔ, < ڮ, ݆ = 1, … , ݊, in [ܽ, ܾ] such that

ஶ

െ ݕ, )൱൲ , ݇ אԳ, ݇ 1. 1 െ ߝ ߚ 1, ݉ = ݈݅݉ ൮ ߚ ൭(ݔ, ՜ஶ

ୀଵ

ୀଵ

ୀଵ

(c) For any ݊ אԳ, ݊ 1, there exists a Borel subset ܤ ܽ[ ؿ, ܾ] such that ݉ = න ݆ ݐିଵ ݀ݐ,

݆ = 1, … , ݊.

(d) For any natural number ݊ 1 and any {ߣଵ , … , ߣ } ؿԹ, the following relation holds true:

ߣ ݉ ߣ (ܾ െ ܽ ). ୀଵ

ୀଵ

The implications (݀) ฺ (ܽ) (ܾ) (ܿ) hold. Remark 4.2. Since ݄ ܮ אஶ ([ܽ, ܾ]) ܮ ؿଶ ([ܽ, ܾ]), we consider the Hilbert base (݁ )ஹଵ obtained by means of Gram-Scmidt procedure applied to the system of polynomials (1,2ݐ, … , ݆ ݐିଵ , … ) = ൫݂ ൯ஹଵ , ݂ ( ݐ݆ = )ݐିଵ for all ݆ { א1,2, … } and all ܽ[ א ݐ, ܾ]. Then we have:

݁ = ߙ, ݂ ฺ< ݄, ݁ > = ߙ, < ݄, ݂ > = ߙ, ݉ , ݊ 1, ୀଵ

ୀଵ

ୀଵ

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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where the coefficients ߙ, are known from Gram-Schmidt procedure. Thus we know all Fourier coefficients < ݄, ݁ >, ݊ 1, in terms of the given moments. Consequently, we know the partial sums of the Fourier series of ݄. If we denote by ( )ஹଵ the sequence of these partial sums, then ԡ݄ െ ԡଶ ՜ 0. On the other hand, for eaxh nonconstant polynomial , we have the following approximation by step-functions: ( )ݐൎ ܿ, ߯, ()ݐ, ܦ, = ൜;ݐ ,

݉ + 1 ݉ < (< )ݐ ൠ 2 2

for suitable natural numbers ݉ , ݎ. Here ܿ, are known in terms of the polynomial . The accuracy of approximation is improved for large ݎand appropriate ݉ (see [68]). Now we write the decomposition of each open subset ܦ, in its connected components: ܦ, = ራ൫ݕ,, , ݔ,, ൯ ฺ ߯, = ߯൫௬,, ,௫,, ൯ .

אԳ

Finally, we obtain:

݉ ൎ න ݆ ݐିଵ ( ݐ݀)ݐൎ ܿ, න ݆ ݐିଵ ߯, (= ݐ݀)ݐ

,

ܿ, ൭ න ݆ ݐିଵ ߯൫௬,,,௫,,൯ (ݐ݀)ݐ൱ = ,

ܿ, ൭൫ݔ,, െ ݕ,, ൯൱ , ݆ אԳ, ݆ 1. ,

Regarding this conclusion as a system of nonlinear equations in the unknowns ݕ,, , ݔ,, , these unknowns can be found in terms of the polynomials . These polynomials can be determined in terms of the moments ݉ .

Chapter One

54

5. Polynomial approximation on unbounded subsets and the Markov moment problem Polynomial approximation recalled below allows proving the existence as well as the uniqueness of the solution of some classical moment problems on unbounded subsets, partially in terms of signatures of quadratic forms. This section is based on references [35], [48], [60-63]. Lemma 5.1 Let ߰: Թା ൌ ሾ0, ሻ ՜ Թା be a continuous function, such that ݈݅݉ ߰ሺݐሻ exists in Թା . Then there is a decreasing sequence ሺ݄ ሻ in the ௧՜

linear hull of the functions ݁ ሺݐሻ ൌ ݁ݔሺെ݇ݐሻ, ݇ אԳ, א ݐሾ0, λሻ, such that ݄ ሺݐሻ ߰ሺݐሻ, ݐ 0, ݈ אԳ ൌ ሼ0,1,2, … ሽ, ݈݄݅݉ ൌ ߰ uniformly on ሾ0, ሻ. There exists a sequence of polynomial functions ሺ ሻאԳ , ݄ ߰, ݈݅݉ ൌ ߰, uniformly on compact subsets of ሾ0, ሻ. In particular, such polynomial approximation holds for nonnegative continuous functions ߰: Թା ՜ Թା , having compact support. Proof. The idea is to consider the sub-algebra ܵመ ൌ ܵ݊ܽሼ݁Ƹ ; ݇ 0ሽ of ܥሺሾ0, ሿሻ, where ሾ0, ሿ is the Alexandroff extension of ሾ0, ሻ, and ݁Ƹ is the continuous extension of ݁ to ሾ0, ሿ, ݁Ƹ ሺሻ ൌ 0, ݇ 1, ݁Ƹ ሺሻ ൌ 1. This sub-algebra clearly separates the points of ሾ0, ሿ and contains the constant functions. According to Stone-Weierstrass theorem, ܵመ is dense in ܥሺሾ0, ሿሻ. It results that any continuous function ߰: Թା ՜ Թା with the property that the limit lim ߰ሺݐሻ exists in Թା , can be uniformly approximated on Թା by ௧՜

elements from ܵ݊ܽሼ݁ ; ݇ 0ሽ. As is well known, when the convergence is uniform, the approximating sequence ሺ݄ ሻ for ߰ can be chosen such that ݄ ሺݐሻ ߰ሺݐሻ 0 for all א ݐሾ0, ሻ. Assume the following equalities hold:

݄ ൌ ߙ, ݁ , ߙ, אԹ, ݈ ൌ 0,1,2, … ୀ

If ߙ, 0, we obtain ߙ, ݁ ߙ, , , where , is a majorizing partial sum of the power series of ݁ , ݁ ൌ lim , , the convergence being uniform ՜

on any compact subset of Թା . If ߙ, ൏ 0, we deduce ߙ, ݁ ൏ ߙ, ݍ, , where ݍ, is a minorizing partial sum of the power series of ݁ ൌ lim , , ՜

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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and the convergence is uniform on compact subsets of the nonnegative semi-axes. Summing as ݇ = 0,1, … , ݉ , one obtains a polynomial ݄ ߰ on Թା . Since the sum defining has a finite number of terms of such partial sums, we conclude: ՜ ߰ uniformly on compact subsets of Թା , as ݈ ՜ . 7KLVHQGVWKHSURRIƑ In applications, the preceding lemma could be useful in order to prove a similar type result for continuous functions defined only on a compact subset ؿ ܭԹା , taking values in Թା . For such a function ߮: ܭ՜ Թା , one denotes by ߮ : Թା ՜ Թା the extension of ߮ which satisfies ߮ ( = )ݐ0 for all א ݐԹା \ܭ. With these notation, from Lemma 3.1 we infer the next result: Lemma 5.2. Let ؿ ܭԹା be a compact subset and ߮: ܭ՜ Թା a continuous function. Then there exists a sequence ( )אԳ of polynomial functions such that ߮ on Թା , | ՜ ߮, ݈ ՜ , uniformly on ܭ. Proof. The idea is to reduce the proof to that of the preceding Lemma 5.1. Namely, we easily construct a continuous extension ߰: Թା ՜ Թା of ߮, having compact support )߰(ݑݏ, ߰ ߮ . Assuming this is done, if ( )אԳ are as in Lemma 5.1, since ՜ ߰ uniformly on the compact ܭand ߰( )ݐ(߮ = )ݐfor all ݐin ܭ, it results the following first conclusion: |ݑݏ ( )ݐെ ߮( |ݑݏ = |)ݐ ( )ݐെ ߰( |)ݐ՜ 0, ݈ ՜ λ. ௧א

௧א

Moreover, according to Lemma 5.1, we have: ߰ ߮ on Թା . This will end the proof. To construct ߰, let ܽ = ݂݅݊ܭ, ܾ = ܭݑݏ, 0 ܽ ܾ < . It is clear that ߮ might have discontinuities at the ends of the intervals representing connected components of [0, )\ܭ. If ߮(ܾ) = 0, then ߮ is continuous at ܾ and on the entire interval [ܾ, ). If ߮(ܾ) > 0, for an arbitrary ߝ > 0, define ߰ on the interval [ܾ, ܾ + ߝ] as the affine function whose graph is the line segment joining the points ൫ܾ, ߮(ܾ)൯ and (ܾ + ߝ, 0), ߰( = )ݐ0 for all ܾ > ݐ+ ߝ, ߰( )ݐ(߮ = )ݐfor all ݐin ܭ. It remains to define ߰ on each bounded connected component of [0, )\ܭ. Let (ݐଵ , ݐଶ ) be such an interval, ݐଵ , ݐଶ ܭ א, ݐଵ < ݐଶ and 0 < ߝ < (ݐଶ െ ݐଵ )Τ2. On the interval [ݐଶ െ ߝ, ݐଶ ] we define ߰ as the affine function whose graph is the line segment of ends (ݐଶ െ ߝ, 0), ൫ݐଶ , ߮(ݐଶ )൯. Similarly, on the interval [ݐଵ , ݐଵ + ߝ], we consider the line segment joining the points ൫ݐଵ , ߮(ݐଵ )൯, (ݐଵ + ߝ, 0). The definition at points ݐଵ , ݐଶ is in accordance with the previous condition and ߰( )ݐ(߮ = )ݐfor all ܭ א ݐ. On the interval (ݐଵ + ߝ, ݐଶ െ ߝ), ߰ ؠ. Finally, if ܽ > 0 and ߮(ܽ) > 0, taking 0 < ߝ < ܽ,

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Chapter One

we define ߰ on the interval [ܽ െ ߝ, ܽ] as being the function whose graph is the line segment joining the points (ܽ െ ߝ, 0), ൫ܽ, ߮(ܽ)൯, ߰ ؠ on [0, ܽ െ ߝ]. If ܽ = 0, we have ߰(ܽ) = ߮(ܽ) since ܽ ܭ אand the interval (0, ܽ) is empty. If ܽ > 0 and ߮(ܽ) = 0, we define ߰ ؠ on [0, ܽ]. Thus ߰ is defined, non-negative, continuous on [0, ), ߰| ߮ ؠ, and )߰(ݑݏis compact, contained in [0, ܾ + ߝ]. The proof is complete. Ƒ Recall that a determinate (Mെdeterminate) measure is uniquely determinate by its moments, or, equivalently, by its values on polynomials. Lemma 5.3. Let ߥ be a M-determinate positive regular measure on [0, λ), with finite moments of all natural orders. If ߰, ( )are as in Lemma 5.1, ଵ then there exists a subsequence ( ~ p lm ) m , such that ՜ ߰ in ܮఔ ([0. λ)) and uniformly on compact subsets. In particular, it follows that the cone ࣪ାା of positive polynomials on [0, λ) is dense in the positive cone (ܮଵఔ ([0. λ)))ା of ܮଵఔ ([0. λ)). Lemma 5.4. (See [48], [58] and [60]]). Let ك ܨԹ be an unbounded closed subset, and ߥ an M-determinate positive regular Borel measure on A, with finite absolute moments of all natural orders. Then for any א ݔ ܥ ()ܨ, )ݐ(ݔ 0, ܵ א ݐ, there exists a sequence ( ) , ݔ, ݉ א Գ, ՜ ݔin ܮଵఔ ()ܨ. In particular, we have lim න ( = ߥ݀)ݐන ߥ݀)ݐ(ݔ,

ி

ி

࣪ା is dense in (ܮଵఔ ())ܨା , and ࣪ is dense in ܮଵఔ ()ܨ. Proof. Let consider the sublattice ܺଵ ܮ ؿଵఔ ( )ܨof all functions ߰ such that |߰| is dominated by some polynomial p on ܵ. To prove the assertions of the statement, it is sufficient to show that for any א ݔ൫ܥ ()ܨ൯ା , we have ܳଵ ( ݂݊݅ ؔ )ݔቊන ;ߥ݀)ݐ( ݔ, ࣪ א ቋ = න ߥ݀)ݐ(ݔ. ி

ி

Obviously, one has ܳଵ ( )ݔ න ߥ݀ )ݐ(ݔ ி

(1.22)

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To prove the converse, we define the linear form ܶ : ܺ ؔ ࣪۩ܵ }ݔ{՜ Թ, ܨ ( + ߙ ؔ )ݔන ߥ݀)ݐ(+ ߙܳଵ ()ݔ, ி

࣪ א , ߙ אԹ. Next we show that ܨ is positive on ܺ . In fact, for ߙ < 0, one has (from the definition of ܳଵ , which is a sublinear functional on ܺଵ ): + ߙ ݔ 0 ฺ െߙ( ฺ ݔെߙ)ܳଵ (ܳ = )ݔଵ (െߙ )ݔ න ߥ݀)ݐ( ி

ฺ ܶ ( + ߙ )ݔ 0. If ܽ 0, we infer that

0 = ܳଵ () = ܳଵ (ߙ ݔെ ߙ )ݔ ߙܳଵ ( )ݔ+ ܳଵ (െߙฺ )ݔ න ߥ݀)ݐ( ܳଵ (െߙ )ݔ െߙܳଵ (ܶ ฺ )ݔ ( + ߙ )ݔ 0. ி

Whence, in both possible cases, we have ݔ ܺ( א )ା ฺ ܶ (ݔ ) 0. Since ܺ contains the space of polynomials functions, which is a majorizing subspace of ܺଵ , there exists a linear positive extension ܶ: ܺ ՜ Թ of ܶ , which is continuous on ܥ ( )ܨwith respect to the sup-norm. Therefore, ܶ has a representation by means of a positive Borel regular measure ߤ on ܨ, such that ܶ( = )ݔන ߤ݀ )ݐ(ݔ, ܥ א ݔ ()ܨ. ி

Let ࣪ א ା be a nonnegative polynomial function. There is a nondecreasing sequence (ݔ ) of continuous nonnegative function with compact support, such that ݔ ա pointwise on ܨ. Positivity of ܶ and Lebesgue dominated convergence theorem for ߤ yield

න )(ܶ = ߥ݀)ݐ( ݔ(ܶݑݏ ) = ݑݏන ݔ ( = ߤ݀ )ݐන ߤ݀)ݐ(, ி

ி

ி

58

Chapter One

࣪ א ା . Thanks to Haviland theorem, there exists a positive Borel regular measure ߣ on ܨ, such that ߣ( )(ߥ = )െ ߤ( ) ߥ( )(ߣ = )+ ߤ(),

࣪ א .

Since ߥ is assumed to be M-determinate, it follows that ߥ( )ܤ(ߣ = )ܤ+ ߤ()ܤ for any Borel subset ܤof ܨ. From this last assertion, approximating each ܮ( א ݔଵఔ ()ܨା , by a nondecreasing sequence of nonnegative simple functions, and also using Lebesgue convergence theorem, one obtains firstly for positive functions, then for arbitrary ߥ -integrable functions ߮: න ߮݀ߥ = න ߮݀ߣ + න ߮݀ߤ , ߮ ܮ אଵఔ ()ܨ. ி

ி

ி

Hence, we must have න ߥ݀ݔ න ܶ = )ݔ(ܶ = ߤ݀ݔ (ܳ = )ݔଵ ()ݔ ி

(1.23)

ி

Now (1.22) and (1.23 FRQFOXGHWKHSURRIƑ The preceding lemma is no longer valid if we replace ܮଵఔ ( )ܨby the Hilbert space ܮଶఔ ()ܨ: there exist ܯെ determinate measures ߥ on F = Թ୬ , n 2 for which the polynomials are not dense in ܮଶఔ (( )ܨsee [6], Theorem 4.4, where the authors construct such a measure Ȟ). Using Bernstein polynomials in several variables, Lemma 5.4 for ݊ = 1 and Fubini’s theorem, one can prove the following result. Lemma 5.5. Let ߥ = ߥଵ × ߥ × ڮ be a product of n ܯെ determinate positive regular Borel measures on Թା = [0, λ), with finite moments of all natural orders. Then we can approximate any nonnegative continuous compactly supported function in ܺ = ܮଵఔ (Թା ) by means of dominating sums of tensor products ଵ ٔ ۪ ڮ , positive polynomial on the real nonnegative semiaxis, in variable ݐ [ א0, λ), ݆ = 1, … , ݊, where: (ଵ ٔ ۪ ڮ )(ݐଵ , … , ݐ ) = ଵ (ݐଵ ) ڮ (ݐ ).

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Notice that a similar result holds for products of M-determinate positive regular measures ߥ on Թ, with finite moments Թ ݐ ݀ߥ of all natural orders ݉ אԳ. The following statement holds for any closed unbounded subset ؿ ܨԹ , hence does not depend on the form of positive polynomials on ܨ. We denote by ߮ , ߮ (ݐ ؔ )ݐଵభ ݐ ڮ , ݆ = (݆ଵ , … , ݆ ) אԳ , ݐ( = ݐଵ , … , ݐ ) ܨ א. Theorem 5.6. Let ܨbe a closed unbounded subset of Թ , ܻ an order complete Banach lattice, ൫ݕ ൯אԳ a given sequence in ܻ, ߥ a positive regular ܯെdeterminate Borel measure on ܨ, with finite moments of all orders. Let ܶଶ ܮ(ܤ אଵఔ (ܵ), ܻ) be a linear positive (bounded) operator from ܮଵఔ ( )ܨto ܻ. The following statements are equivalent: (a) there exists a unique linear operator ܶ ܤ א൫ ܮଵ,ఔ ()ܨ, ܻ൯ such that ܶ൫߮ ൯ = ݕ , ݆ אԳ , ܨis between and ܶଶ on the positive cone of ܮଵఔ ()ܨ, and ԡܶԡ ԡܶଶ ԡ; (b) for any finite subset ܬ ؿԳ , and any ൛ܽ ൟא ؿԹ, we have బ

σאబ ܽ ߮ 0 on ฺ ܨ0 σאబ ܽ ݕ σאబ ܽ ܶଶ ൫߮ ൯. We go on by recalling a result on the form of non-negative polynomials in a strip [34], which leads to a simple solution for the related Markov moment problem. Theorem 5.7. Suppose that ݐ(ଵ , ݐଶ ) אԹ[ݐଵ , ݐଶ ] is non – negative on the strip [ = ܣ0,1] × Թ. Then ݐ(ଵ , ݐଶ ) is expressible as ݐ(ଵ , ݐଶ ) = ߪ(ݐଵ , ݐଶ ) + ߬(ݐଵ , ݐଶ )ݐଵ (1 െ ݐଵ ), where ߪ(ݐଵ , ݐଶ ), ߬(ݐଵ , ݐଶ ) are sums of squares in Թ[ݐଵ , ݐଶ ]. Let [ = ܨ0,1] × Թ, ߥ a positive ܯെ determinate regular Borel measure on ܨ, with finite moments of all orders, ܺ = ܮଵఔ ()ܨ, ߮ (ݐଵ , ݐଶ ) ؔ ݐଵభ ݐଶమ , ݆ = (݆ଵ , ݆ଶ ) אԳଶ , (ݐଵ , ݐଶ ) ܨ א. Let ܻ be on order complete Banach lattice, ൫ݕ ൯אԳమ a sequence of given elements in ܻ. The next result follows as a consequence of Theorems 5.6 and 5.7.

60

Chapter One

Theorem 5.8. Let ܶଶ ܤ אା (ܺ, ܻ) be a linear (bounded) positive operator from ܺ to ܻ. The following statements are equivalent: (a) there exists a unique (bounded) linear operator ܶ: ܺ ՜ ܻ such that ܶ൫߮ ൯ = ݕ , ݆ אԳଶ , ܶ is between zero and ܶଶ on the positive cone of ܺ, ԡܶԡ ԡܶଶ ԡ; (b) for any finite subset ܬ ؿԳଶ , and any ൛ߣ ; ݆ ܬ א ൟ ؿԹ, we have

0 ߣ ߣ ݕା ߣ ߣ ܶଶ ൫߮ା ൯; ,אబ

,אబ

0 ߣ ߣ ൫ݕభ ାభାଵ,మାమ െ ݕభାభାଶ,మ ାమ ൯ ,אబ

ߣ ߣ ቀܶଶ ൫߮భାభାଵ,మ ାమ െ ߮భାభାଶ,మ ାమ ൯ቁ, ,אబ

݅ = (݅ଵ , ݅ଶ ) ܬ א , ݆ = (݆ଵ , ݆ଶ ) ܬ א .

Theorem 5.9. Let ܺ be as in Lemma 5.5, ൫ݕ ൯אԳ a sequence in ܻ, where ܻ is an order complete Banach lattice and let ܶଶ ܤ אା (ܺ, ܻ) be a positive bounded linear operator. The following statements are equivalent: (a) there exists a unique (bounded) linear operator ܶ ܺ(ܤ א, ܻ) such that, ܶ൫߮ ൯ = ݕ , ݆ אԳ , ܶ is between zero and ܶଶ on the positive cone of X,ԡܶԡ ԡܶଶ ԡ; (b) for any finite subset ܬ ؿԳ and any ൛ߣ ; ݆ ܬ א ൟ ؿԹ, we have ߣ ߮ ( )ݐ 0 א ݐԹା ฺ ߣ ݕ ܻ אା ; אబ

אబ

for any finite subsets ܬ ؿԳ, ݇ = 1, … , ݊, and any ൛ߣೖ ൟ

ೖ אೖ

1, … , ݊, the following relations hold

ؿԹ, ݇ =

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ቌ ڮቌ ߣభ ߣభ … ߣ ߣ ݕభ ାభାభ,..,, ା ା ቍ ڮቍ భ ,భ אభ

భ ,భ אభ

, א

ቌ ڮቌ ߣభ ߣభ … ߣ ߣ ܶଶ (߮భାభାభ ,..,, ାା ቍ ڮቍ , ,א

(݈ଵ , … , ݈ ) { א0,1} .

From Theorem 5.9 written for ݊ = 1, also using the explicit form of nonnegative polynomials on [0, λ), we obtain the following: Corollary 5.10. Let ߥ be a ܯെ determinate positive regular Borel measure on Թା = [0, λ), with finite moments of all natural orders, ܺ = ܮଵఔ (Թା ). Let ܻ be an order complete Banach lattice, ܶଶ א ܤା (ܺ, ܻ), ൫ݕ ൯אԳ a sequence in ܻ. The following statements are equivalent: (a) there exists a unique (bounded) linear operator ܶ ܺ(ܤ א, ܻ) which verifies T൫߮ ൯ = ݕ , ݆ אԳ, ܶ ܶଶ on ܺା , ԡܶԡ ԡܶଶ ԡ; (b) for any finite subset ܬ ؿԳ and any ൛ߣ ; ݆ ܬ א ൟ ؿԹ, the following inequalities hold true 0 ߣ ߣ ݕାା ߣ ߣ ܶଶ ൫߮ାା ൯, ݇ { א0,1}. ,אబ

,אబ

We recall that ࣪ א , )ݐ( 0 for all א ݐԹା ݍ = )ݐ(ଶ ( )ݐ+ ݎݐଶ ()ݐ א ݐԹା , for some ݍ, ࣪ א ݎand )ݐ( 0 for all אԹ ݍ = )ݐ(ଶ ( )ݐ+ ݎଶ ( א ݐ )ݐԹ, for some ݍ, ࣪ א ݎ. Similarly to Theorem 5.9, denoting

߮ (ݐ ؔ )ݐଵభ ݐ ڮ , ݆ = (݆ଵ , … , ݆ ) אԳ , ݐ( = ݐଵ , … , ݐ ) אԹ the following result holds true:

62

Chapter One

Theorem 5.11. Let ܺ = ܮଵఔ (Թ ), where ߥ = ߥଵ × ߥ × ڮ is a product of ݊ ܯെ determinate positive regular Borel measure on Թ, with finite moments of all orders, ܻ be an order complete Banach lattice, ൫ݕ ൯אԳ a sequence in ܻ, ܶଶ ܤ אା (ܺ, ܻ). The following statements are equivalent: (a) there exists a unique (bounded) linear operator ܶ ܺ(ܤ א, ܻ) such that ܶ൫߮ ൯ = ݕ , ݆ אԳ , ܶ is between zero and ܶଶ on the positive cone of X, ԡܶԡ ԡܶଶ ԡ; (b) for any finite subset ܬ ؿԳ and any ൛ߣ ; ݆ ܬ א ൟ ؿԹ, we have ߣ ߮ ( )ݐ 0 א ݐԹ ฺ ߣ ݕ ܻ אା , אబ

אబ

for any finite subsets ܬ ؿԳ, ݇ = 1, … , ݊, and any ൛ߣೖ ൟ

ೖ אೖ

ؿԹ, ݇ =

1, … , ݊, the following relations hold

ቌ… ቌ ߣభ ߣభ … ߣ ߣ ݕభାభ,..,, ା ቍ ڮቍ భ ,భ אభ

, א

ቌ… ቌ ߣభ ߣభ … ߣ ߣ ܶଶ (߮భାభ,..,, ା ቍ ڮቍ . భ ,భ אభ

,א

Corollary 5.12. Let consider the hypothesis and notations from Corollary 5.10, where we replace Թା with Թ. The following statements are equivalent: (a) there exists a unique (bounded) linear operator ܶ ܺ(ܤ א, ܻ) which verifies ܶ൫߮ ൯ = ݕ , ݆ אԳ, ܶ ܶଶ on ܺା , ԡܶԡ ԡܶଶ ԡ; (b) for any finite subset ܬ ؿԳ and any ൛ߣ ; ݆ ܬ א ൟ ؿԹ, the following inequalities hold true 0 ߣ ߣ ݕା ߣ ߣ ܶଶ ൫߮ା ൯. ,אబ

,אబ

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The next result is not necessarily related to the moment problem. However, it is characterizing the positivity of a bounded linear operator in terms of quadratic forms only. From this viewpoint, one solves the difficulty arising from the fact that nonnegative polynomials on Թ , ݊ 2, might not be expressible as sums of squares. Corollary 5.13. Let ܺ be as in Theorem 5.11, and ܻ be a Banach lattice. Assume that ܶ is a linear bounded operator from ܺ to ܻ. The following statements are equivalent: (a) ܶ on the positive cone of ܺ; (b) for any finite subsets ܬ ؿԳ, ݇ = 1, … , ݊, and any ൛ߣೖ ൟ

ೖ אೖ

ؿ

Թ, ݇ = 1, … , ݊, the following relations hold:

0 ൮ ڮ൮ ߣభ ߣభ ߣ ڮ ߣ ܶ൫ݔభାభ , … , ା ൯൲ ڮ൲. భ ,భ אభ

. א

Proof. Notably, statement (b) claims that ܶ is positive on the convex cone generated by special positive polynomials similar to those mentioned in Lemma 5.5, each factor of any term in the sum being nonnegative on the entire real axis. Consequently, (a)֜(b) is obvious. In order to prove the converse, observe that any nonnegative element of ܺ can be approximated by nonnegative continuous compactly supported functions. Such functions can be approximated by sums of tensor products of positive polynomials in each separate variable. The conclusion is that any nonnegative function from ܺ can be approximated in ܺ = ܮଵఔ (Թ ) by sums of tensor products of squares of polynomials in each separate variable. But on such special polynomials, ܶ admits nonnegative values, following the condition (b). Now the desired conclusion is a consequence of the continuity of ܶ. This HQGVWKHSURRIƑ Next, we consider a more general case, when the sandwich condition 0 ܶ ܶଶ on the positive cone of the domain space is replaced with the condition ܶଵ ܶ ܶଶ on the positive cone ܺା under attention. Here ܶଵ , ܶଶ are two given linear operators satisfying some conditions. A part of these conditions are expressible in terms of quadratic forms (with vector or scalar coefficients). Let ؿ ܭԹା be an arbitrary compact subset. We denote by ܺ = )ܭ(ܥthe Banach lattice of all real valued continuous functions on ܭ and let ܻ be an arbitrary order complete Banach lattice. One denotes

64

Chapter One

߮ ( ݐ = )ݐ , א ݐԹା , ݆ אԳ. Theorem 5.14. Let ܶଵ , ܶଶ be two linear operators from ܺ to ܻ, such that ܶଵ ܶଶ on the positive cone of ܺ, and (ݕ )ஹ a given sequence of elements in ܻ. The following statements are pairwise equivalent: (a) there exists a unique (bounded) linear operator ܶ: ܺ ՜ ܻ such that ܶ൫߮ ൯ = ݕ , ݆ אԳ, ܶଵ ܶ ܶଶ on the positive cone of ܺ, ԡܶଵ ԡ ԡܶԡ ԡܶଶ ԡ; (b) for any polynomial σ ୀ ߙ ߮ on ܭ, it results σୀ ߙ ܶଵ ൫߮ ൯ σ ୀ ߙ ݕ ; if ܬ ؿԳ is a finite subset and ൛ߣ ; ݆ ܬ א ൟ ؿԹ, then ߣ ߣ ݕାା ߣ ߣ ܶଶ ൫߮ାା ൯, ݈ { א0,1}; ,אబ

,אబ

(c) ܶଵ ܶଶ on ܺା and for any polynomial σאబ ߣ ߮ , the following inequality holds: ା

ି

ߣ ݕ ܶଶ ቌቌ ߣ ߮ ቍ ቍ െ ܶଵ ቌቌ ߣ ߮ ቍ ቍ. אబ

אబ

אబ

Proof. According to notations and assertions of (a), the implication (ܽ) ฺ (ܾ) is obvious. To prove the converse implication, we observe that first assertion of (b) says that defining

ܶ ቆ

ୀ

ߙ ߮ ቇ =

ୀ

ߙ ݕ , ݉ אԳ, ߙ אԹ,

we obtain a linear operator defined on the subspace of polynomial functions, which verifies the moment conditions ܶ ൫߮ ൯ = ݕ , ݆ אԳ, and ܶ ܶଵ on the convex cone ࣪ା of all polynomial functions which are nonnegative on ܭ. On the other hand, any element from ܺ = )ܭ(ܥis dominated by a constant function, so that the subspace ࣪ of polynomial functions defined on Թା verifies the hypothesis of Theorem 2.22, where ܺଵ stands for ܺ, and ܺ stands for ࣪. According to Theorem 2.22, the linear operator ܶ െ ܶଵ : ࣪ ՜ ܻ, which is positive on ࣪ା = ࣪ ܺ תା admits a

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positive linear extension ܷ: ܺ ՜ ܻ. We define ܶ = ܶଵ + ܷ ܶଵ on ܺା . In addition ܶ ܮ אା (ܺ, ܻ) verifies ܶ൫߮ ൯ = ܶଵ ൫߮ ൯ + ܷ൫߮ ൯ = ܶଵ ൫߮ ൯ + ܶ ൫߮ ൯ െ ܶଵ ൫߮ ൯ = ܶ ൫߮ ൯ = ݕ , ݆ אԳ. In other words, ܶ: ܺ ՜ ܻ is a linear extension of ܶ : ࣪ ՜ ܻ, which dominates ܶଵ on ܺା . Next we prove that ܶ ܶଶ on ܺା . To this end, observe that according to the second assertion of (b), we already know that ܶ ܶଶ on special polynomial functions which are nonnegative on the entire semi axes Թା . Indeed, any nonnegative polynomial )ݐ( = on Թା has the explicit form ݍ = )ݐ(ଶ ( )ݐ+ ݎݐଶ ( )ݐfor some ݍ, א ݎԹ[]ݐ. On the other hand, since ܶ ܶଵ , the linear operator ܶ is positive and hence is also continuous; ܶଶ is continuous as well, thanks to its positivity. We now apply Lemma 5.2 for an arbitrary ߮ ܺ אା . Using the notations of Lemma 5.2 and the above discussed assertions we infer that ܶଵ (߮) ܶ(߮) = lim ܶ( ) lim ܶଶ ( ) = ܶଶ (߮), ߮ ܺ אା . ՜

՜

It remains to prove the last relation of (a). If ߰ is an arbitrary function in ܺ, then ܶ(߰) ܶ(|߰|) ܶଶ (|߰|) and similarly െܶ(߰) = ܶ(െ߰) ܶଶ (|߰|). These inequalities yield |ܶ(߰)| ܶଶ (|߰|) and, since ܻ is a Banach lattice, the conclusion is ԡܶ(߰)ԡ ԡܶଶ (|߰|)ԡ ԡܶଶ ԡԡ|߰|ԡ = ԡܶଶ ԡԡ߰ԡ, ߰ in ܺ. Thus ԡܶԡ ԡܶଶ ԡ. Similarly, ԡܶଵ ԡ ԡܶԡ. The equivalence (ܽ) (ܿ) follows directly from Theorem 2.32. This completes the proof. Ƒ Remark 5.1. It would be useful to know whether a similar result to that of Lemma 5.1 holds when we replace Թା by Թ. In this case the dominating polynomials should be nonnegative on the entire real axes and hence would be sums of squares. If ߰: Թ ՜ Թା would be continuous, compactly supported and even function, then the problem is reduced to that solved by Lemma 5.1: ߰ can be approximated by dominating even polynomials, the convergence holding uniformly on compact subsets of Թ. If ߰ is not even, while the other assumptions on it are maintained, the polynomial approximation is not obvious.

66

Chapter One

Remark 5.2. In Theorem 5.14, the main implication is (ܾ) ฺ (ܽ), since conditions of (b) are checkable in terms of the moments ݕ and the given operators ܶଵ , ܶଶ . Remark 5.3. It would be interesting to prove results such as Lemmas 5.1 and 5.2 in several variables. Namely, being given a nonnegative continuous compactly supported real function ݂ defined on Թା , ݊ 2, and denoting by ܭits support, we could approximate ݂ on ܭ ك ܭଵ × ܭ × ڮ , where ܭ = ݎ ()ܭ, ݆ = 1, … , ݊, by sums of products ݂ଵ ٔ ݂ ٔ ڮ , ݂ : ܭ ՜ Թା being continuous for all ݆ = 1, … , ݊, via Stone-Weierstrass theorem or Bernstein polynomials of ݊ variables. Then we could apply to each ݂ Lemma 5.2, and finally obtain approximation of ݂ by finite sums of products of polynomials ଵ ٔ ٔ ڮ , where : Թା ՜ Թା , ݂ on ܭ , ݆ = 1, … , ݊, and the approximation holds uniformly on compact subsets of Թା . The motivation for such consideration is that being given a system of commuting positive self-adjoint operators ܣଵ , … , ܣ acting on a Hilbert space, we could try to consider polynomial uniform approximation on = ܭ ߪ(ܣଵ ) × ܣ(ߪ × ڮଵ ), where ߪ൫ܣ ൯ is the spectrum of ܣ , ݆ = 1, … , ݊, in order to solve related Markov moment problems over ܭ. If ك ܨԹ is an arbitrary closed unbounded subset, then we denote by ࣪ା the convex cone of all polynomial functions (with real coefficients), taking nonnegative values at any point of ܨ. ࣪ାା will be a sub-cone of ࣪ା generated by special nonnegative polynomials expressible in terms of sums of squares. One denotes by ܥ ( )ܨthe vector space of real valued continuous and compactly supported functions defined on ܨ. Theorem 5.15. Let ك ܨԹ be a closed unbounded subset, ߥ a positive Borel moment determinate measure on ܨ, having finite moments of all orders, ܺ = ܮଵఔ ()ܨ, ߮ ( ݐ = )ݐ , ܨ א ݐ, ݆ אԳ . Let ܻ be an order complete Banach lattice, ൫ݕ ൯אԳ a given sequence of elements in ܻ, ܶଵ and ܶଶ two bounded linear operators from ܺ to ܻ. Assume that there exists a sub – cone ࣪ାା ࣪ كା such that each ݂ א൫ܥ ()ܨ൯ା can be approximated in ܺ by a sequence ( ) , ࣪ אାା , ݂ for all ݈. The following statements are equivalent: (a) there exists a unique (bounded) linear operator ܶ: ܺ ՜ ܻ, ܶ൫߮ ൯ = ݕ , ݆ אԳ , ܶଵ ܶ ܶଶ on ܺା , ԡܶଵ ԡ ԡܶԡ ԡܶଶ ԡ;

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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(b) for any finite subset ܬ ؿԳ and any ൛ߣ ; ݆ ܬ א ൟ ؿԹ, the following implications hold true: ߣ ߮ ࣪ אା ฺ ߣ ܶଵ ൫߮ ൯ ߣ ݕ , אబ

ߣ ߮ ࣪ אାା ฺ ߣ ܶଵ ൫߮ ൯ , ߣ ݕ ߣ ܶଶ ൫߮ ൯. אబ

(1.24)

אబ

אబ

אబ

אబ

(1.25)

אబ

Proof. We start by observing that the first condition (1.25) implies the positivity of the bounded linear operator ܶଵ , via its continuity. Indeed, if ݂ א ൫ܥ (ܵ)൯ା , ࣪ אାା , ݂ for all ݈, ՜ ݂ in ܮଵఔ ()ܨ, then, according to the first condition (1.25), ܶଵ ( ) for all ݈ אԳ and the continuity of ܶଵ yields ܶଵ (݂) = lim ܶଵ ( ) .

Since ൫ܥ ()ܨ൯ା is dense in ܺା via measure theory reasons, the continuity of ܶଵ implies ܶଵ on ܺା . Thus ܶଵ is a positive linear operator. Next, we define ܶ : ࣪ ՜ ܻ, ܶ ൫σאబ ߣ ߮ ൯ = σאబ ߣ ݕ , where the sums are finite and the coefficients ߣ are arbitrary real numbers. Condition (1.24) says that ܶ െ ܶଵ on ࣪ା . If we consider the vector subspace ܺଵ of ܺ formed by all functions ߰ ܺ אhaving the modulus |߰| dominated by a polynomial א ࣪ା on the entire set ܨ, then ࣪ is a majorizing subspace of ܺଵ and ܶ െ ܶଵ is a positive linear operator on ࣪. Application of Theorem 2.22 leads to the existence of a positive linear extension ܷ: ܺଵ ՜ ܻ, of ܶ െ ܶଵ . Obviously, ܺଵ contains ܥ ( )ܨ+ ࣪: ൫ = || ฺ ࣪ א ඥ ή ଶ ( + ଶ )Τ2 ࣪ א൯. Indeed, since ߮ ܥ א ( א |߮| ฺ )ܨ൫ܥ ()ܨ൯ା ฺ |߮| ܾ ( ࣪ אaccording to Weierstrass’ Theorem), we infer that ߮ ܺ אଵ ; here ܾ < λ is a real number. Hence ܥ (ܺ ؿ )ܨଵ . Now let ݓ ;࣪ א e observe that: 1 + ଶ െ 2|( = |1 െ |)|ଶ can be written as | |

+ ଶ ࣪ א. 2

68

Chapter One

According to the definition of ܺଵ , it results ࣪ ܺ ؿଵ . Consequently ܥ ( )ܨ+ ࣪ ܺ ؿଵ . Going back to the positive linear extension ܷ: ܺଵ ՜ ܻ, of ܶ െ ܶଵ , we conclude that ܶ = ܷ + ܶଵ : ܺଵ ՜ ܻ is an extension of ܶ , ܶ ܶଵ on (ܺଵ )ା , and ܶ (ܶ = ) ( ) ܶଶ ( )for all ࣪ א ାା , according to the last requirement (1.25). A first conclusion is: ܶଵ ( ) ܶ ( ) ܶଶ ( )for all ࣪ א ାା , ߰ ܺ( אଵ )ା . ܶ (߰) ܶଵ (߰) ,

(1.26)

Our next goal is to prove continuity of ܶ on ܥ ()ܨ. Let (݂ )ஹ be a sequence of nonnegative continuous compactly supported functions, such that ݂ ՜ in ܺଵ , and a sequence of polynomials ݂ , ࣪ אାା for all ݈, such that the following convergence result holds: ԡ െ ݂ ԡଵ ՜ 0, ݈ ՜ λ. Then ԡ ԡଵ ԡ െ ݂ ԡଵ + ԡ݂ ԡଵ ՜ 0, ݈ ՜ λ. Now (1.26) and the continuity of ܶଵ , ܶଶ , yield: ՚ ܶଵ ( ) ܶ ( ) ܶଶ ( ) ՜ , hence ܶ ( ) ՜ . It results: ܶଵ (݂ ) ܶ (݂ ) ܶ ( ) ՜ , hence ܶ (݂ ) ՜ 0. If (݃ )ஹ is and arbitrary sequence of compactly supported and continuous functions such that ݃ ՜ in ܺଵ , then ݃ା ՜ , ݃ି ՜ . According to what we already have proved, we can write ܶ (݃ା ) ՜ and ܶ (݃ି ) ՜ , which further yield ܶ (݃ ) ՜ . This proves the continuity of ܶ on ܥ ()ܨ, and the this subspace ܥ ( )ܨis dense in ܺ. Hence there exists a unique continuous linear extension ܶ ܺ(ܤ א, ܻ) of ܶ . It results ܶଵ ܶ ܶଶ on ܺା , ԡܶଵ ԡ ԡܶԡ ԡܶଶ ԡ, ܶ൫߮ ൯ = ܶ ൫߮ ൯ = ݕ , ݆ אԳ .

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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Indeed, ܶଵ , ܶ, ܶଶ are linear, continuous, and ࣪ାା is dense in ൫ܥ ()ܨ൯ା , hence is dense in ܺା as well. For an arbitrary ߮ ܺ א, the following inequalities hold true, via the preceding remarks: ±ܶ(߮) = ܶ(±߮) ܶ(|߮|) ܶଶ (|߮|) ฺ |ܶ(߮)| ܶଶ (|߮|) ฺ ԡܶ(߮)ԡ ԡܶଶ (|߮|)ԡ ԡܶଶ ԡԡ߮ԡ. The conclusion is: ԡܶԡ ԡܶଶ ԡ, and similarly, ԡܶଵ ԡ ԡܶԡ. The uniqueness of the solution ܶ follows by the density of polynomials in ܺ, via continuity of the linear operator ܶ. 7KLVHQGVWKHSURRIƑ Our next goal is to give a result for Markov moment problem in the space ܮଵఔ (Թ), where ߥ is a moment-determinate measure on Թ, having finite moments Թ ݐ ݀ߥ of all orders ݇ אԳ. Corollary 5.16. Let ܺ = ܮଵఔ (Թ), where ߥ is a moment determinate positive Borel measure on Թ, with finite moments of all orders. Assume that ܻ is an arbitrary order complete Banach lattice, and (ݕ )ஹ is a given sequence having its terms in ܻ. Let ܶଵ , ܶଶ be two linear operators from ܺ to ܻ such that ܶଵ ܶଶ on ܺା . The following statements are equivalent: (a) there exists a unique bounded linear operator ܶ from ܺ to ܻ, ܶଵ ܶ ܶଶ on ܺା , ԡܶଵ ԡ ԡܶԡ ԡܶଶ ԡ, such that ܶ(߮ ) = ݕ for all ݊ אԳ; (b) if ܬ ؿԳ is a finite subset and ൛ߣ ; ݆ ܬ א ൟ ؿԹ, then

ߣ ߣ ܶଵ ൫߮ା ൯ ߣ ߣ ݕା ߣ ߣ ܶଶ ൫߮ା ൯. ,אబ

,אబ

,אబ

Corollary 5.17. Let ߥ be a moment determinate positive Borel measure on Թ, with finite moments of all orders. Assume that ݄ଵ , ݄ଶ are two functions in ܮஶ ఔ (Թ), such that ݄ଵ ݄ଶ almost everywhere. Let (ݕ )ஹ be a given sequence of real numbers. The following statements are equivalent: (a) there

exists

݄ ܮ אஶ ఔ (Թ), such

that

݄ଵ ݄ ݄ଶ ߥ െ almost

everywhere, Թ ݕ = ߥ݀)ݐ(݄ ݐ for all ݆ אԳ; (b) if ܬ ؿԳ is a finite subset and ൛ߣ ; ݆ ܬ א ൟ ؿԹ, then

70

Chapter One

ߣ ߣ න ݐା ݄ଵ ( ߥ݀)ݐ ߣ ߣ ݕା ,אబ

Թ

,אబ

ߣ ߣ න ݐା ݄ଶ (ߥ݀)ݐ. ,אబ

Թ

In cases of Corollaries 5.16, 5.17, we have ࣪ାା = ࣪ା . Going further to the multidimensional case, for examples of such sub-cones ࣪ାା of ࣪ା and their applications to the Markov moment problem see [60]. Namely, these theorems emphasize the importance of using quadratic forms in the multidimensional case when nonnegative polynomials are not usually expressible as sums of squares. In both these examples, the inclusion ࣪ାା ؿ ࣪ା is strict. In the case of = ܨԹା (respectively = ܨԹ ), ݊ 2, the cone ࣪ାା consists in all polynomials which are sums of products of the form ଵ ۪ ۪ ڮ , where each , ݆ = 1, … , ݊, is a nonnegative polynomial on Թା (respectively on Թ), hence is expressible by means of sums of squares of polynomials of one variable. Proceeding this way, the last conditions of (1.25) can be written in terms of quadratic forms (see Corollary 5.18 stated below). Corollary 5.18. Let ߥ = ߥଵ × ߥ × ڮ , ݊ 2, ߥ being an ܯെdeterminate (moment-determinate) positive regular Borel measure on Թ, ݆ = 1, … , ݊, ܺ = ܮଵఔ (Թ ), ߮ ( ݐ = )ݐ , א ݐԹ , ݆ אԳ . Additionally assume that ߥ has finite moments of all orders, ݆ = 1, … , ݊. Let ܻ be an order complete Banach lattice, ൫ݕ ൯אԳ a given sequence of elements in ܻ, ܶଵ and ܶଶ two bounded linear operators from ܺ to ܻ. The following statements are equivalent: (a) there exists a unique (bounded) linear operator ܶ: ܺ ՜ ܻ, ܶ൫߮ ൯ = ݕ , ݆ אԳ , ܶଵ ܶ ܶଶ on ܺା , ԡܶଵ ԡ ԡܶԡ ԡܶଶ ԡ; (b) for any finite subset ܬ ؿԳ and any ൛ߣ ; ݆ ܬ א ൟ ؿԹ, the following implication holds true:

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

71

ߣ ߮ ࣪ אା ฺ ߣ ܶଵ ൫߮ ൯ ߣ ݕ ; אబ

אబ

אబ

for any finite subsets ܬ ؿԳ, ݇ = 1, … , ݊, and any ൛ߣೖ ൟ

ೖ אೖ

ؿԹ, the

following inequalities hold true:

൮ ڮ൮ ߣభ ߣభ ߣ ڮ ߣ ܶଵ ൫߮భ ାభ,..,, ା ൯൲ ڮ൲ ; భ ,భ אభ

. א

ቌ ڮቌ ߣభ ߣభ ߣ ڮ ߣ ݕభ ାభ ,..,, ା ቍ ڮቍ భ ,భ אభ

. א

൮ ڮ൮ ߣభ ߣభ ߣ ڮ ߣ ܶଶ ൫߮భାభ,..,, ା ൯൲ ڮ൲ . భ ,భ אభ

. א

The last consequences of Theorem 5.15 refer to Markov moment problem on Թା . Corollary 5.19. Let ܺ = ܮଵఔ (Թା ), where ߥ is a moment determinate positive Borel measure on Թା , with finite moments of all orders. Assume that ܻ is an arbitrary order complete Banach lattice, and (ݕ )ஹ is a given sequence having its terms in ܻ. Let ܶଵ , ܶଶ be two linear operators from ܺ to ܻ such that ܶଵ ܶଶ on ܺା . As usual, we denote ߮ ( ݐ = )ݐ , ݆ אԳ, א ݐԹା . The following statements are equivalent: (a) there exists a unique bounded linear operator ܶ from ܺ to ܻ, ܶଵ ܶ ܶଶ on ܺା , ԡܶଵ ԡ ԡܶԡ ԡܶଶ ԡ, such that ܶ(߮ ) = ݕ for all ݊ אԳ; (b) if ܬ ؿԳ is a finite subset and ൛ߣ ; ݆ ܬ א ൟ ؿԹ, then

ߣ ߣ ܶଵ ൫߮ାା ൯ ߣ ߣ ݕାା ߣ ߣ ܶଶ ൫߮ାା ൯, ,אబ

,אబ

݇ { א0,1}.

,אబ

72

Chapter One

Corollary 5.20. Let ߥ be a moment determinate positive Borel measure on Թା , with finite moments of all orders. Assume that ݄ଵ , ݄ଶ are two functions in ܮஶ ఔ (Թା ), such that ݄ଵ ݄ଶ almost everywhere. Let (ݕ )ஹ be a given sequence of real numbers. The following statements are equivalent: (a) there

exists

݄ ܮ אஶ ఔ (Թା ), such

that

݄ଵ ݄ ݄ଶ ߥ െ almost

everywhere, Թ ݕ = ߥ݀)ݐ(݄ ݐ for all ݆ אԳ; శ

(b) if ܬ ؿԳ is a finite subset and ൛ߣ ; ݆ ܬ א ൟ ؿԹ, then: ߣ ߣ න ݐାା ݄ଵ ( ߥ݀)ݐ ߣ ߣ ݕାା Թశ

,אబ

,אబ

ߣ ߣ න ݐାା ݄ଶ (ߥ݀)ݐ, ݇ { א0,1}. ,אబ

Թశ

Corollary 5.21. Assume that ݄ଵ is a function in ܮஶ ௫(ି௧)ௗ௧ (Թା ), such that ݄ଵ almost everywhere. Let (ݕ )ஹ be a given sequence of real numbers. The following statements are equivalent: (a) there exists ݄ ܮ אஶ ௫(ି௧)ௗ௧ (Թା ), such that ݄ଵ ݄ almost everywhere, Թ ݐ ݄(ݕ = ߥ݀)ݐ for all ݆ אԳ; శ

(b) if ܬ ؿԳ is a finite subset and ൛ߣ ; ݆ ܬ א ൟ ؿԹ, then: ߣ ߣ න ݐାା ݄ଵ (ି ݁)ݐ௧ ݀ ݐ ߣ ߣ ݕାା ,אబ

Թశ

,אబ

ߣ ߣ (݅ + ݆ + ݇)!, ݇ { א0,1}. ,אబ

Proof. We apply Corollary 5. 20 to ݀ߥ = ݁(ݔെݐ݀)ݐ. According to [76, Theorem 2, p. 498], the measure ݁(ݔെ ݐ݀)ݐis moment determinate. As it is well-known from elementary properties of Gamma function, its moments RIDOORUGHUVDUHILQLWH7KHFRQFOXVLRQIROORZVƑ

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

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6. A constrained optimization problem related to Markov moment problem The present section has as motivation proving similar results to some of those of [41]. One proves a result in a general setting, obtained by means of Theorem 2.32 stated above. A constrained related optimization problem in infinite dimensional spaces is solved too. The results presented in the sequel were published in [53]. Using the latter theorem, one obtains a necessary and sufficient condition for the existence of a feasible solution (see theorem 6.1 from below). Under such condition, the existence of an optimal feasible solution follows too. On the other hand, the uniqueness and the construction of the optimal solution seems to be not obtained easily by such general methods. Therefore, we focus mainly on the existence problem. For other aspects of such problems on an optimal solution (uniqueness or non – uniqueness, construction of a unique solution, etc.), see [41]. In the latter work, one considers the following primal problem (P): study the constrained minimization problem: ߥ = ݂݅݊ ቊԡ߮ԡ ; ߮ ܮ אఓ (ܼ), න ݂߮ ݀ߤ = ܾ , ݆ = 1, … , ݊, ߙ ߮ ߚቋ

where ߙ, ߚ are in

ܮஶ ఓ (ܼ), ൫݂ ൯ୀଵ is a subset of

ܮଵఓ (ܼ), and ܾ =

(ܾଵ , … , ܾ )௧ אԹ . The function ߮ is unknown, and in general it is not determined by a finite number of moments. The next theorem discusses some of the above existence type results for a feasible solution. Here (ܼ, ࣧ) is a measure space endowed with a ߪ െ finite positive measure ߤ, and ࣧ is the ߪ െ algebra of all measurable subsets of ܼ. Theorem 6.1. Let ( א 1, λ) and let q be the conjugate of p. Let ൫݂ ൯א be

an arbitrary family of functions in ܮఓ (ܼ), where the measure ߤ is ߪ – finite, and ൫ܾ ൯א a family of real numbers. Assume that ߙ, ߚ ܮ אఓ (ܼ) are such that ߙ ߚ.. The following statements are equivalent:

(a) there exists ߮ ܮ אఓ (ܼ) such that ݂߮ ݀ߤ = ܾ , ܬ א ݆ , ߙ ߮ ߚ; (b) for any finite subset ܬ ܬ كand any ൛ߣ ൟא ؿԹ, the following బ

implication holds:

74

Chapter One

ߣ ݂ = ߰ଶ െ ߰ଵ , ߰ଵ , ߰ଶ אቀ ܮఓ (ܼ)ቁ ฺ ߣ ܾ ା

אబ

אబ

න ߚ߰ଶ ݀ߤ െ න ߙ߰ଵ ݀ߤ;

Moreover, the set of all feasible solutions ߮ (satisfying the conditions (a)) is weakly compact with respect the dual pair (ܮ , ܮ ) and the inferior

ߥ = ݂݅݊ ቊԡ߮ԡ ; ߮ ܮ אఓ (ܼ), න ݂߮ ݀ߤ = ܾ , ݆ ܬ א, ߙ ߮ ߚቋ

ԡߙԡ is attained for at least one optimal feasible solution ߮ . Proof. Since the implication (ܽ) ฺ (ܾ) is obvious, the next step consists in proving that (ܾ) ฺ (ܽ). We define the linear positive (continuous) forms ܶଵ , ܶଶ on ܺ = ܮఓ (ܼ), by ܶଵ (݂) = ߙ݂݀ߤ, ܶଶ (߮) = ߚ݂݀ߤ, ݂ ܺ א. Then condition (b) of the present theorem coincides with condition (b) of Theorem 2.32. A straightforward application of the latter theorem, leads to the existence of a linear form ܶ on ܺ, such that the interpolation conditions ܶ൫߮ ൯ = ܾ , ݆ ܬ א, are verified and න ߙ߰݀ߤ ܶ(߰) න ߚ߰݀ߤ , ߰ ܺ אା .

In particular, the linear form ܶ is positive on ܺ = ܮఓ (ܼ), and this space is a Banach lattice. It is known that on such spaces, any linear positive functional is continuous (see [37], [39], [69] or other sources). The conclusion is that ܶ can be represented by means of a nonnegative function ߮ ܮ אఓ (ܼ). From the previous relations, we infer that න ߙ߰݀ߤ න ߮߰݀ߤ න ߚ߰݀ߤ , ߰ ܺ אା .

Writing these relations for ߰ = ߯ , where ܤis an arbitrary measurable set of positive measure ߤ()ܤ, one deduces

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

න (߮ െ ߙ) ݀ߤ 0, න (ߚ െ ߮) ݀ߤ 0,

75

ࣧ א ܤ, ߤ( > )ܤ0.

Now a standard measure theory argument shows that ߙ ߮ ߚ almost everywhere in ܼ. This finishes the proof of (ܾ) ฺ (ܽ). To prove the last assertion of the theorem, observe that the set of all feasible solutions is weakly compact in ܮఓ (ܼ), by Alaoglu’s theorem (it is a weekly closed subset of the closed ball centered at the origin, of radius ԡߚԡ , and ܮఓ (ܼ) is reflexive). On the other hand, the norm of any normed linear space is lower weakly semi – continuous, as the supremum of continuous linear forms, which are also weak continuous with respect to the dual pair ቀܮఓ (ܼ), ܮఓ (ܼ)ቁ , 1 < < λ, 1Τ + 1Τ = ݍ1. Since ܮఓ (ܼ) is reflexive for 1 < < ݍλ, we conclude that the norm ԡήԡ is weakly lower semicontinuous on the weakly (convex) and compact set described at point (a), so that it attains its minimum at a function ߮ of this set. Hence, there exists at least one optimal feasible sROXWLRQ7KLVFRQFOXGHVWKHSURRIƑ

Remark 6.1. If the set ൛݂ ൟא is total in the space ܮఓ (ܼ), then the set of all feasible solutions is a singleton, so that there exists a unique solution. Remark 6.2. Theorem 6.1 seems to cannot be deduced from the preceding proof type arguments for = 1, = ݍ. Indeed, the closed unit ball of ܮஶ ఓ (ܼ) is weakly compact, but with respect to the weak topology associated ஶ to the pair ൫ܮଵఓ (ܼ), ܮஶ ఓ (ܼ) ൯. On the other hand, the usual norm on ܮఓ (ܼ) is lower semi-continuous with respect to the weak topology associated to the dual pair څ

څ

ቀܮఓ (ܼ), ቀܮఓ (ܼ)ቁ ቁ and ቀܮఓ (ܼ)ቁ = ቀܮଵఓ (ܼ)ቁ not reflexive.

څڅ

് ܮଵఓ (ܼ), since ܮଵఓ (ܼ) is

Remark 6.3. In the proof of Theorem 6.1 we claimed that any positive linear functional on ܮఓ (ܼ), 1 < < λ, is continuous. Actually, there is a much more general result on this subject. Namely, any positive linear operator acting on ordered Banach spaces is continuous (see [37] or/and [39] for the proof).

76

Chapter One

7. A remark on Riemann series In the classical one-dimensional moment problem setting, the moments are defined as integrals of natural powers of the variable. However, the following integral makes sense not only with respect to Lebesgue measure ݀ݐ, but also as an integral in the sense of Riemann. Namely, consider the integral ଵ

න ݐఈ ݀ = ݐ1Τ(ߙ + 1) , ߙ > 0.

(1.27)

Since the power function under the integral sign is continuous on [0,1], it is Riemann integrable and (1.27) holds via well-known Leibniz-Newton formula. Using this first remark, we can write the integral (11) as the limit of the associated Riemann integral sums, choosing an appropriate divisionpoints in [0,1]. The result of this subsection is stated as follows: Theorem 7.1. The following evaluations hold true: 1 + 2ఈ + ڮ+ ݊ఈ ~ ݊ఈାଵ Τ(ߙ + 1) , ݊ ՜ , ߙ > 0, 1 + 1Τ2ఈ + ڮ+ 1Τ݊ఈ ~ ݊ଵିఈ /(1 െ ߙ), ݊ ՜ , 0 ߙ < 1.

(1.28) (1.29)

Proof. We consider the equidistant points ݐ = ݆Τ݊ , ݊ { א1,2, … }, ݆ א {1, … , ݊}; then οݐ = ݐ െ ݐିଵ = 1Τ݊ for all ݆ { א1, … , ݊}. Hence max οݐ = 1Τ݊ ՜ 0. According to the remark preceding the ଵஸஸ

statement, for any positive real number we conclude: ଵ

1Τ(ߙ + 1) = න ݐఈ ݀= ݐ

= lim ݐఈ οݐ ՜

ୀଵ

= lim (1Τ݊)(1/݊ఈ + 2ఈ /݊ఈ + ڮ+ ݊ఈ /݊ఈ ) ՜

On Hahn-Banach Type Theorems, Polynomial Approximation on Unbounded Subsets, the Moment Problem and Mazur-Orlicz Theorem

= lim ቆ ՜

77

(1 + 2ఈ + ڮ+ ݊ఈ ) ቇ , ߙ > 0. ݊ఈାଵ

Hence (1.28) is proved. In other words, 1Τ(ߙ + 1) is the limit of the arithmetic means of 1Τ݊ఈ , 2ఈ Τ݊ఈ , ڮ. ݊ఈ Τ݊ఈ , as ݊ ՜ λ. To prove (1.29), we compute the following limit, representing the inverse (with respect to multiplication of numbers) of the limit of harmonic mean for the sequence appearing in (1.30) from below: ߚ(ߙ) = lim (1 + 1Τ2ఈ + ڮ+ 1Τ݊ఈ )Τ݊ଵିఈ , 0 < ߙ < 1. ՜

(1.30)

Applying the Stolz-Cesàro test for limits of the type λΤλ, we are led to the following computation: 1 (݊ + 1)ఈ ߚ(ߙ) = lim = ՜ஶ (݊ + 1)ଵିఈ െ ݊ଵିఈ 1 1 ݊ + 1 ݔ + 1 lim = lim . ՜ஶ ௫՜ஶ ݊ ଵିఈ ݔଵିఈ ൬1 െ ቀ ቁ ൰ ൬1 െ ቀ ቁ ൰ ݊+1 ݔ+1 The last limit is of the type 0Τ0, and all the involved functions are VPRRWK$SSO\LQJO¶+۾SLWDO¶VUXOHZHREWDLQ 1 ( ݔ+ 1)ଶ = 1Τ(1 െ ߙ). ߚ(ߙ) = lim 1 ି ݔఈ ௫՜ஶ െ(1 െ ߙ) ቀ ቁ ݔ+1 ( ݔ+ 1)ଶ െ

Thus (1.29) is also proved. This ends the proof. Ƒ

CHAPTER TWO ELEMENTS OF REPRESENTATION THEORY

In the present chapter, we follow results and ideas of [5], [38], [65]. We start by an interpretation of Carathéodory’s Theorem 2.6 as an integral representation theorem (by means of a discrete measure). Then, using Krein-Milman Theorem 2.17 and a passing to the limit procedure (eventually involving convergent subnets), one obtains integral representations in terms of arbitrary probability measures. In what follows ܭis a compact convex nonempty subset of a (Hausdorff) locally convex space ܧ. For ܭ א ݕ, one denotes by ߜ௬ the “point mass” at ݕ, that is ߜ௬ is the Borel measure which equals 1 on any Borel subset of ܭwhich contains ݕ, and equals 0 otherwise. According to these comments, if ܭ א ݔand ܭis contained in an ݊ െ dimensional subspace of ܧ, there exist ݁ଵ , … , ݁ାଵ ାଵ ߙ = 1, such that = ݔ extreme points of ܭand ߙଵ , … , ߙାଵ in Թା , σୀଵ ାଵ ାଵ σୀଵ ߙ ݁ . Let ߤ = σୀଵ ߙ ߜೕ . Then for any continuous linear form ܮon ܧ, one obtains: ାଵ

ߜ௫ ( = )ݔ(ܮ = )ܮ

ୀଵ

ାଵ

ߙ ܮ൫݁ ൯ =

= න ߤ݀ܮ.

ୀଵ

ߙ ߜೕ ()ܮ(ߤ = )ܮ (2)

This last assertion is what we mean when we say that ߤ represents ݔ. In the sequel, a probability measure on ܭmeans a nonnegative regular Borel measure ߤ on ܭ, with ߤ( = )ܭ1. Definition 1. Suppose that ܭis a nonempty compact subset of a locally convex space ܺ and ߤ is a probability measure on ܭ. A point ݔin ܺ is said to be represented by ߤ if = )ݔ(ܮ ߤ݀ܮ

Elements of Representation Theory

79

for every continuous linear functional ܮon ܺ (other terminology: “ ݔis the barycenter of ߤ", “ ݔis the resultant of ߤ”). Note that any point ܭ א ݔis trivially represented by ߜ௫ ; the interesting fact pointed out by (2) is that for a convex compact subset ܭof a finite dimensional space, each ݔin ܭmay be represented by a probability measure which “is supported” by the extreme points of ܭ. A similar result holds for arbitrary convex compact metrizable subsets ܭof ܺ (see Theorem 3.1 below). Definition 2. If ߤ is a nonnegative regular Borel measure on the compact Hausdorff space ܭand ܤis a Borel subset of ܭ, we say that ߤ is supported by ܤif ߤ( = )ܤ\ܭ0. Theorem 1. (Choquet). Suppose that ܭis a metrizable compact convex subset of the locally convex space ܺ, and that ݔ is an element of ܭ. Then there is a probability measure ߤ on ܭ, which represents ݔ and is supported by the extreme points of ܭ. For the proof of the preceding theorem, see [65, p. 14-15]. The next result is somehow similar to Choquet’s theorem, without requiring metrizability condition on ( ܭsee [2, p. 17]). Theorem 2. (Choquet-Bishop-de Leeuw). Suppose that ܭis a compact convex subset of the locally convex space ܺ, and that ݔ is in ܭ. Then there is a probability measure ߤ on ܭwhich represents ݔ and which vanishes on every Baire subset of ܭwhich is disjoint from the set of extreme points of ܭ. We recall two density results related to the theorems reviewed in this chapter. Let ܭbe compact convex subset of the locally convex space ܺ. We denote by )ܭ(ܥthe Banach lattice of all real valued continuous functions on ܭ. )ܭ(ݒ݊ܥwill be the convex cone of all convex continuous real valued functions on ܭ, and ת )ܭ(ݒ݊ܥ = )ܭ(ܣ൫െ)ܭ(ݒ݊ܥ൯ the vector space of all continuous affine functions on ܭ. Finally, we denote: ܺ | څ + Թ ή1={| څ ݔ + ߙ; څ ܺ א څ ݔ, ߙ אԹ}. Lemma 3. (i) )ܭ(ݒ݊ܥെ )ܭ(ݒ݊ܥis a dense subspace of )ܭ(ܥ.

80

Chapter Two

(ii) ܺ | څ + Թ ή is a dense subspace of )ܭ(ܣand the inclusion ܺ | څ + Թ ή )ܭ(ܣ ؿmay be strict. For the proof of Lemma 3, see [38, p. 308-309]. Theorems 1 and 2 claim that any point in ܭis the barycenter of a probability measure essentially defined by its behavior on the set of extreme points of ܭ. The following question arises naturally: does any probability measure on ܭhave a barycenter? The answer is affirmative, and, moreover, for a given probability measure ߤ on ܭ, there exists a unique corresponding barycenter, denoted ܾܽ)ߤ(ݎ. Namely, the following result holds: Theorem 4. (See [38, Lemma 7.2.3, p. 310]). If ܭis a compact convex subset in the locally convex space ܺ and ߤ is a probability measure on ܭ, there exists a unique point ܾܽ ܭ א )ߤ(ݎsuch that ܮ൫ܾܽ)ߤ(ݎ൯ = න ߤ݀ܮ

for all continuous linear functionals ܮon ܺ. Since all the locally convex spaces are assumed to be Hausdorff, the uniqueness of ܾܽ )ߤ(ݎfollows from the fact that the topological dual ܺ څof ܺ separates the points of ܺ. The next result follows from the more general Theorem 7.2.4 of [38] and represents the Jensen integral inequality for a barycenter and for probability measures. Theorem 5. (Jensen; see [38]). Suppose that ߤ is a probability measure on the convex compact subset ܭof the locally convex space ܺ. Then ݂൫ܾܽ)ߤ(ݎ൯ න ݂()ݔ(ߤ݀)ݔ

for all continuous convex functions ݂: ܭ՜ Թ. Theorems 4 and 5 are valid in the more general case when ߤ is a SteffensenPopoviciu measure, which goes beyond the framework of nonnegative measures (see [38]). Next, we recall some results on the uniqueness of the representing measure. The uniqueness holds if and only if the compact convex subset ܭis a simplex. Before going to infinite dimensional simplexes, we review the definition of a finite dimensional simplex. The sets of the form = ܥ

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81

ܿݔ{( , … , ݔே }) are called polytopes. If ݔଵ െ ݔ , … , ݔே െ ݔ are linearly independent, then ܥis called an ܰ െsimplex, with vertices ݔ , … , ݔே . In this case, ݀݅݉ ܰ = ܥand any point ݔof ܥhas a unique representation as a convex combination of vertices, ே

=ݔ

ୀ

ே

ߙ ݔ , ߙ אԹା = [0, λ),

ୀ

ߙ = 1.

The numbers ߙ , … , ߙே are called the barycentric coordinates of ݔ. The standard ܰ െsimplex (or unit ܰ െsimplex) in Թேାଵ is defined by: οே = ൜(ߙ , … , ߙே ) אԹேାଵ ;

ே

ߙ = 1, ߙ 0, ݇ = 0, … , ܰൠ.

ୀ

Now we go on with infinite dimensional simplexes. As is shown in [65, p. 51-52], for studying a compact convex subset ܭof a locally convex space ܺ and see when ܭis a simplex, it is easier to assume that ܭis a base of a convex cone ( ܥwith vertex at the origin), i. e. ܥ ؿ ܭand ܥ א ݕif and only if there exists a unique ߙ 0 and ݔin ܭsuch that ݔߙ = ݕ. Moreover, as discussed in [65, p.52], whenever a compact convex subset is a base for a cone ܥ, we can always assume that it is of the form ܥ ת ܪfor some closed hyperplane ܪin ܺ which misses the origin. Definition 3. If a convex set ( ܭnot necessarily compact) is a base of a cone ෩, we call ܭa simplex if the space ܭ ෩െܭ ෩ is a vector lattice in the ordering ܭ ෩ induced by ܭ. Definition 4. Let ܺ ؿ ܭbe a compact convex subset; if ߥ and ߤ are nonnegative regular Borel measures on ܭ, we write ߥ ߤ ظif ߥ(݂) ߤ(݂) for all continuous convex functions on ܭ. Lemma 6. (See [65, p. 18]). If ߥ is a nonnegative measure on ܭ, then there exists a maximal measure ߤ such that ߤ ߥ ظ. For the complete statement of the next result see [65, p. 56-57]. Theorem 7. (Choquet-Meyer). Suppose that ܭis a non-empty compact convex subset of the locally convex space ܺ. Then ܭis a simplex if and only if for each point ݔin ܭthere is a unique maximal measure ߤ௫ on ܭsuch that ߤ௫ (݄) = ݄( )ݔfor all continuous affine functions ݄: ܭ՜ Թ.

CHAPTER THREE EARLIER AND RECENT RESULTS ON CONVEXITY AND OPTIMIZATION

1. Introduction Recall that solving optimization problems for convex (and concave) mappings is simpler than solving similar problems for arbitrary mappings, thanks to using Hahn-Banach type results, which are strongly related to convexity. Therefore, our results refer only to convex optimization problems. For very recent results on subdifferentials of convex operators see [39] and the references therein.

2. On convex functions defined on bounded convex subsets of Թ In this section, we improve the basic result from [43] (see also the references there). Theorem 2.1. Let X be an arbitrary real vector space, ܺ ؿ ܤa finite dimensional convex bounded subset, Y an order complete vector lattice and ߮: ܤ՜ ܻ a convex operator. Then there exists ݕ ܻ אsuch that ߮( )ݔ ݕ for all ܤ א ݔ. Proof. Since B is finite dimensional and convex, its relative interior )ܤ(݅ݎ is nonempty. Recall that by )ܤ(݅ݎone denotes the interior of B with respect to the topology induced on ܤby that on the (finite dimensional) linear variety generated by B. As is well known, ߮ is subdifferentiable at any point of )ܤ(݅ݎ. Let ܾ )ܤ(݅ݎ אand ܶ a translation of a subgradient of ߮ at ܾ , that is an affine operator ܶ: ܺ ՜ ܻ such that ܶ(ݔ ) = ߮(ݔ ) and ܶ( )ݔ ߮()ݔ for all ܤ א ݔ. On the other hand, let ݔଵ , … , ݔାଵ be (at most) + 1 affine independent points in the linear variety generated by B, such that ܿ ك ܤ൛ݔଵ , … , ݔାଵ ൟ >

Earlier and Recent Results on Convexity and Optimization

83

Here p is the linear dimension of the linear variety generated by B. Such a system of points does exist thanks to the fact that B is finite dimensional and bounded. Now the following relations hold ାଵ

ାଵ

߮( )ݔ ܶ( ܶ = )ݔቌ ߙ ݔ ቍ = ߙ ܶ൫ݔ ൯ ୀଵ

ୀଵ

ାଵ

ቌ ߙ ቍ ݂݅݊൛ܶ൫ݔ ൯; 1 ݆ + 1ൟ = ୀଵ

݂݅݊൛ܶ൫ݔ ൯; 1 ݆ + 1ൟ ؔ ݕ , ାଵ ାଵ where = ݔσୀଵ ߙ ݔ , ߙ 0, ݆ = 1, … , + 1, σୀଵ ߙ = 1.

This concludes the proof. Ƒ Remark 2.1. The preceding proof furnishes a constructive method of finding a lower bound ݕ for ߮()ܤ. In the next theorem, we show that the only convex subsets ܺ ؿ ܤfor which any convex real function on ܤis bounded below, are the finite dimensional convex bounded subsets. Theorem 2.2. Let ܺ be an arbitrary real infinite dimensional vector space and ܺ ؿ ܤa convex subset, such that any convex real function defined on B is bounded from below. Then B is contained in a finite dimensional subspace of X and is bounded there. Proof. Let څ ݔbe an arbitrary linear functional in the algebraic dual ܺ څof X. Then څ ݔand െ څ ݔare convex, and, by hypothesis, both of them are bounded from below on B. Thus )ܤ( څ ݔis bounded in Թ. Hence B is weakly bounded in ܺ, endowed with the weak topology corresponding to the dual pair (ܺ, ܺ ) څ. Let us endow X with the finest locally convex topology which is compatible with this dual pair. By [69], we derive that B is bounded in the latter topology. Application of [69, exercise 7, Chapter II, p. 69], leads to the fact that B is contained in a finite dimensional subspace and bounded WKHUH7KLVFRQFOXGHVWKHSURRIƑ

84

Chapter Three

3. Elements of minimum norm and related results In the sequel, we present the characterization of an element of minimum norm in terms of linear continuous forms of norm one, related to the distance function, in arbitrary normed linear spaces. Related geometric aspects are briefly discussed. Recall that any closed hyperplane in a normed linear space is the level set of a continuous linear form. Lemma 3.1. Let ܺ be a real normed vector space, }ߙ = ܶ{ = ܪa closed hyperplane in ܺ and ݔ ܺ א. Then the distance ݀(ݔ , )ܪis given by formula: ݀(ݔ , = )ܪ

|்(௫బ )ିఈ| ԡ்ԡ

.

(3.1)

Theorem 3.2. Let X be a real normed vector space, ܺ ؿ ܣa closed convex subset not containing the origin and ܽ ܣ א. The following statements are equivalent: (a) ԡܽ ԡ = ݂݅݊{ԡܽԡ; ܽ ;}ܣ א (b) there exists ܶ څ ܺ א, such that ԡܶԡ = 1, ԡܽ ԡ = ܶ(ܽ ) ܶ(ܽ) ;ܣ א ܽ (c) there exists a closed homogeneous hyperplane ܪsuch that ݀(ܪ, = )ܣԡܽ ԡ. Proof. Let B be the open ball of radius ԡܽ ԡ, centered at the origin. From (a) we infer that the intersection of ܤwith ܣis empty. From the separation Hahn-Banach theorem, we infer that there exists a hyperplane, = ܪ {ܶ = ߙ}, ܶ {\ څ ܺ א0}, which separates ܤand ܣ. Scaling by a suitable constant, we can assume that ߙ = ԡܽ ԡ. Hence, we have: < ܶ{ ؿ ܤԡܽ ԡ}, ܶ{ ؿ ܣ ԡܽ ԡ}

(3.2)

If ԡݔԡ < 1, then we can write ฮԡܽ ԡݔฮ < ԡܽ ԡ ฺ ܶ(ԡܽ ԡ < )ݔԡܽ ԡ. This leads to ԡܶԡ 1. Since ܽ ܣ א, from the second inclusion (3.2) we infer that

Earlier and Recent Results on Convexity and Optimization

ܶ(ܽ ) ԡܽ ԡ ฺ ܶ ൬

85

ܽ ൰ 1 ฺ ԡܶԡ 1. ԡܽ ԡ

The conclusion is: ԡܶԡ = 1, ԡܽ ԡ ܶ(ܽ ) ԡܶԡ ή ԡܽ ԡ = ԡܽ ԡ ฺ ܶ(ܽ) ԡܽ ԡ = ܶ(ܽ ) ܣ א ܽ, where we have used (3.2) once more. Now, the proof of (a) ฺ(b) is complete. (b) ฺ(c). Let ܶ be a linear continuous functional verifying (b), ି ܶ = ܪଵ ({0}). From this and also using Lemma 3.1, one obtains for any ܣ א: ݀(ܽ, = )ܪ

|ܶ(ܽ) െ 0| = ܶ(ܽ) ԡܽ ԡ, ܣ(݀ ฺ ܣ א ܽ. )ܪ ԡܽ ԡ. ԡܶԡ

On the other hand, ݀(ܽ, )ܪ ݀(ܽ , ܽ(ܶ = )ܪ ) = ԡܽ ԡ. Comparing the preceding relations, we conclude that ݀(ܪ, = )ܣԡܽ ԡ. (c) ฺ(a). This implication is almost obvious: ԡܽ ԡ = ݀(ܪ, )ܣ ݀(0, ฺ )ܣԡܽ ԡ = ݂݅݊{ԡܽԡ; ܽ }ܣ א. TKHSURRIRIWKHWKHRUHPLVFRPSOHWHƑ Corollary 3.3. Let ܺ be a real Hilbert space, ܺ ؿ ܣa closed convex subset not containing the origin and ܽ ܣ א. The following statements are equivalent: (a) ԡܽ ԡ = ݂݅݊{ԡܽԡ; ܽ ;}ܣ א (b) ԡܽ ԡଶ < ܽ , ܽ > ;ܣ א ܽ (c) ݀({ܽ }ୄ , = )ܣԡܽ ԡ; Proof. (a) ฺ(b). From the corresponding implication of Theorem 3.1, there exists ܶ څ ܺ אsuch that

86

Chapter Three

ԡܶԡ = 1, ԡܽ ԡ = ܶ(ܽ ) ܶ(ܽ) ฺ ܣ א ܽ ܺ א ݑ, ܶ =< ݑ,ή>, ԡݑԡ = 1, < ݑ, ܽ > = ԡܽ ԡ = ԡܽ ԡ ή ԡݑԡ. It follows that in Cauchy-Schwarz-Buniakovski inequality equality occurs, so that we must have: ܽߣ = ݑ ฺ ԡܽ ԡ = < ݑ, ܽ > = ߣ < ܽ , ܽ > < ݑ, ܽ ߣ < ܽ , ܽ > ฺ ߣ=

1 , ԡܽ ԡଶ < ܽ , ܽ > ;ܣ א ܽ ԡܽ ԡ

The implication (b) ฺ(c) follows from the corresponding implication of Theorem 3.2, taking ܽ{ = ܪ }ୄ = {}ݑୄ . The implication (c) ฺ(a) also follows from the corresponding implication of the preceding theorem, applied for the same hyperplane ܽ{ = ܪ }ୄ Ƒ We recall the following related earlier results. Theorem 3.4. Let ܺ be a normed vector space, ܺ ؿ ܥa closed convex cone, ݔ ܥ\ܺ א,

݀ = ݀(ݔ , )ܥ.

Then there exists a linear continuous functional ܶ څ ܺ א, such that ԡܶԡ 1, ܶ( )ݑ 0 ܥ א ݑ, ܶ(ݔ ) = ݀ . Proof. Let ܲ: ܺ ՜ Թ be the functional defined by ܲ(ݔ(݀ = )ݔ, {݂݊݅ = )ܥԡ ݔെ ݕԡ; }ܥ א ݕ, ܺ א ݔ. Obviously, we have ܲ(ݔ ) = ݀ , We prove that ܲ is sublinear: 1 ߣ > 0 ฺ ܲ(ߣ{݂݊݅ = )ݔԡߣ ݔെ ݕԡ; ݂݊݅ = }ܥ א ݕ൜ฯߣ ݔെ ߣ ݕฯ ; ܥ א ݕൠ ߣ = ߣ ݂݅݊{ԡ ݔെ ݑԡ; )ݔ(ܲߣ = }ܥ א ݑ, ܺ א ݔ. For ߣ = 0, one obtains ܲ(ߣ(ܲ = )ݔ0) = 0 = ߣܲ()ݔ. On the other hand, for all ݔଵ , ݔଶ ܺ א, ݕଵ , ݕଶ ܥ א,,one has:

Earlier and Recent Results on Convexity and Optimization

87

ԡݔଵ െ ݕଵ ԡ + ԡݔଶ െ ݕଶ ԡ ԡ(ݔଵ + ݔଶ ) െ (ݕଵ + ݕଶ )ԡ ܲ(ݔଵ + ݔଶ ) ฺ ݂݅݊{ԡݔଵ െ ݕଵ ԡ; ݕଵ }ܥ א+ ݂݅݊{ԡݔଶ െ ݕଶ ԡ; ݕଶ }ܥ א ܲ(ݔଵ + ݔଶ ) ฺ ܲ(ݔଵ + ݔଶ ) ܲ(ݔଵ ) + ܲ(ݔଶ ) ݔଵ , ݔଶ ܺ א. Let ܺ = {ߙݔ ; ߙ אԹ}, ܶ : ܺ ՜ Թ, ܶ (ߙݔ ) = ߙ݀ , א ߙԹ. Then ܶ is linear and for ߙ 0 we have ܶ (ߙݔ ) = ߙ݀ = ߙܲ(ݔ ) = ܲ(ߙݔ ), while for all ߙ < 0, the following remark holds ܶ (ߙݔ ) = ߙ݀ < 0 ܲ(ߙݔ ). The conclusion is that ܶ is bounded from above by ܲ on the onedimensional subspace ܺ . According to the Hahn-Banach theorem, we infer that there exists a linear extension ܶ: ܺ ՜ Թ of ܶ , such that ܶ ܲ on ܺ. These yields: ܶ( )ݕ 0 ܥ א ݕ, ܶ( )ݔ ܲ( )ݔ ݀(ݔ, ) = ԡݔԡ ܺ א ݔ, ܶ(ݔ ) = ݀ = ݀(ݔ , ݔ(ܲ = )ܥ ). Thus ܶ is continuous, of norm at most 1, and satisfies all the other assertion from the statement. The proof is complete. Ƒ Remark 3.1. If ܺ is a reflexive Banach space, then the distance ݀(ݔ , )ܥ is attained at least at one point ݕ of ܥ. In this case, we have ܶ(ݔ ) = ԡݔ െ ݕ ԡ ฺ ܶ(ݔ െ ݕ ) ܶ(ݔ ) = ԡݔ െ ݕ ԡ ฺ ܶ൬

ݔ െ ݕ ൰ 1 ฺ ԡܶԡ = 1. ԡݔ െ ݕ ԡ

In particular, in any Hilbert space, the linear functional of Theorem 3.2 is of norm 1. Theorem 3.5. (See [46]). Let ܺ be a real normed vector space, ܣ, ܤconvex subsets of ܺ such that ݀ = ݀(ܣ, > )ܤ0. Then there exist two closed parallel hyperplanes ܪଵ , ܪଶ which separate the two convex subsets, such that ݀(ܪଵ , ܪଶ ) = ݀(ܣ, )ܤ. Proof. From the hypothesis we derive

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Chapter Three

݀ = ݀(, ܣെ (ܤ ฺ )ܤ, ݀ ) ܣ( תെ ฺ = )ܤ څ ܺ א ܶ, ܶ(݀ < )ݔ (ܤ א ݔ, ݀ ), ܶ( )ݕ ݀ ܣ א ݕെ ܤ. The preceding relations further yield: ԡܶԡ 1, inf ܶ( ߙ ؔ )ܣ ݀ + sup ܶ(݀ ؔ )ܤ + ߚ ฺ ߙ െ ߚ ݀ . Now the closed hyperplanes we are looking for are ܪଵ = {ܶ = ߙ}, ܪଶ = {ܶ = ߚ} From the preceding notations we infer that ܶ{ ك ܣ ߙ}, ܶ{ ك ܤ ߚ}. Since the two hyperplanes separate the convex sets ܣand ܤ, we have ݀(ܣ, )ܤ ݀(ܪଵ , ܪଶ ). In order to prove the converse inequality, observe that the distance between two parallel hyperplanes equals the distance from a point situated on a hyperplane, to the other hyperplane. This last remark, relation ԡܶԡ 1 and Lemma 3.1 formula (3.1), lead to: ݀(ܪଵ , ܪଶ ) = ݀(݄ଵ , ܪଶ ) =

|ܶ(݄ଵ ) െ ߚ| ߙ െ ߚ = ߙ െ ߚ ݀ = ݀(ܣ, )ܤ. ԡܶԡ ԡܶԡ

This concludes the proof. Ƒ In Hilbert spaces, if the algebraic difference ܣെ ܤis closed, then the distance from the preceding theorem is attained at a pair of points (ܽ, ܾ) א ܤ × ܣ. If the two sets have smooth boundaries, the line joining these points is orthogonal to the tangent hyperplanes at these points. These remarks lead to the following result, which is useful in applications. It avoids using Lagrange multipliers in determining the distance between two suitable convex sets, and the attaining points. Corollary 3.6. Let X be a real Hilbert space, ܲ: ܺ ՜ Թ convex and smooth, ܳ: ܺ ՜ Թ concave and smooth, such that ܳ(ܺ א ݔ )ݔ(ܲ < )ݔ. If there exist ܽଵ , ܾଵ ܺ אsuch that

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݀ ቀ൫ܽଵ , ܲ(ܽଵ )൯, ൫ܾଵ , ܳ(ܾଵ )൯ቁ = ݀(݃ܲ ݄ܽݎ, ݃ > )ܳ ݄ܽݎ0 Then there exists ߙ אԹ, s.t. ߘܲ(ܽଵ ) = ߙߘܳ(ܾଵ ), א ߣԹ: (ߘܲ(ܽଵ ), െ1) = ߣ(ܽଵ െ ܾଵ , ܲ(ܽଵ ) െ ܳ(ܾଵ ) ) Examples. 1) As it is well-known (see [68]), if ܪis a Hilbert space, ܪ ؿ ܥ is a closed convex subset such that ܥ ב, then there exists a unique element ܽ ܥ אsuch that ԡܽԡ = inf ԡݔԡ. ௫א

In other words, the distance ݀(, )ܥis attained at a unique point ܽ ܥ א. In particular, this assertion holds true for any closed non-homogeneous hyperplane in ܪ. 2) Let ܺ = ݈ଵ (respectively ܮଵ ([0,1])), ݔ( = ݔ{ = ܤ ) ݈ אଵ ; ݔ 0 א ݊Գ, σஶ ୀ ݔ = 1}, respectively ଵ

= ܤቊ߮ ܮ( אଵ )ା ; න ߮( = ݐ݀)ݐ1ቋ.

In both these cases we have ݀(, = )ܤԡݕԡଵ = 1 ܤ א ݕ. and obviously B is bounded, convex and closed. Hence there are infinitely many points (all the points of B) at which the distance to origin is attained. This happens because of the form of the balls in ݈ଵ - norm (respectively in ܮଵ െ norm).

4. Classes of concave mappings, related constrained inequalities and optimization 4.1. Introduction Using minimum principle for concave functions, we prove a related constrained inequality, firstly for finite sums. The case of infinite sums is

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Chapter Three

deduced from the previously mentioned inequality, passing to the limit. In the end, a similar result for a class of concave operators taking values in the positive cone of a special space of self-adjoint operators is discussed. Two related types of examples are given. We follow the results of [54].

4.2. An inequality related to a special class of concave functions Let ݊ 2 be a natural number, {ܿଵ , … , ܿ } ( ؿ0, λ), ݄: [1, λ) ՜ Թା a concave continuous strictly increasing function, such that ݄(1) = 0. Theorem 4.2.1. The following statements are valid. (a) For any {ݔଵ , … , ݔ } ؿԹା , σୀଵ ݔ = > ܯ0, the following relation holds true

ܿ ݄൫1 + ݔ ൯ ൬ ݉݅݊ ܿ ൰ ݄(1 + )ܯ ୀଵ

(3.3)

ଵஸஸ

If ݄ is strictly concave, then in (3.3) equality occurs if and only if ݆ א {1, … , ݊} is such that ݔ = 0, ݆ { א1, … , ݊}\{݆ }, ݔబ = ܯand ܿబ = ݉݅݊ ܿ . ଵஸஸ

(b) Using the same notations und hypothesis, under the weaker constraint σୀଵ ݔ > ܯ0, for strictly concave and increasing function ݄, the same relation (3.3) holds, and equality occurs in the same case as that mentioned at point (a). Proof. (a) Define the ݊ െ 1 dimensional simplex

ܵିଵ,ெ ؔ ቊ(ݔଵ , … , ݔ ) אԹ ; ݔ 0, ݆ { א1, … , ݊},

ୀଵ

ݔ = ܯቋ

This is a simplex of vertices ݁ଵ ؔ (ܯ, 0, … ,0), … , ݁ ؔ (0,0, … ,0, ( )ܯthe set ݔܧ൫ܵିଵ,ெ ൯ equals the set {݁ଵ , … , ݁ }). ܵିଵ,ெ is contained in the ݊ െ 1 dimensional linear variety ܸିଵ = ൛݁భ ൟ + ܵ ݊ܽቄ݁ െ ݁భ ; ݆ { א1, … , ݊}\{݆ଵ }ቅ for any ݆ଵ { א1, … , ݊}. On the other hand, the function

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݃(ݔ(݃ = )ݔଵ , … , ݔ ) ؔ ܿ ݄൫1 + ݔ ൯, ܵ ؔ ܭ א ݔିଵ,ெ ୀଵ

is concave and continuous, as a sum of ݊ functions having these two properties. Application of the minimum principle for ݃ leads to ݃( )ݔ min ݃൫݁ ൯ = min ቀܿ ݄(1 + )ܯቁ = ൬ ݉݅݊ ܿ ൰ ݄(1 + )ܯ. ଵஸஸ

ଵஸஸ

ଵஸஸ

This proves the first assertion of the theorem. Assume now that ݄ is strictly concave. Then so is ݃. If equality occurs in (3.3) for \ܭ א ݔቄ݁ ; ݆ א {1, … , ݊}ቅ, then, due to Carathéodory’s theorem, there exist a subset ܧଵ = ൛݁భ , … , ݁ೖ ൟ, ݇ { א1, … , ݊} and {ߙଵ , … , ߙ } ( ؿ0, λ), σୀଵ ߙ = 1, such that = ݔσୀଵ ߙ ݁ . Now strict concavity of ݃ and equality in (3.3) yield

݃( > )ݔ ߙ ݃൫݁ ൯ ቀ min ݃൫݁ ൯ቁ ൭ ߙ ൱ min ݃൫݁ ൯ = ଵஸஸ

ୀଵ

ଵஸஸ

ୀଵ

൬ min ܿ ൰ ݄(1 + )ݔ(݃ = )ܯ, ଵஸஸ

which is a contradiction. This concludes the proof the proof of (a). To prove (b), consider the set

ܣெ ؔ ቊ(ݔଵ , … , ݔ ) אԹ ; ݔ 0, ݆ { א1, … , ݊},

ୀଵ

ݔ ܯቋ

= ራ ܵିଵ,ெାఌ . ఌஹ

Then application of the results of (a) leads to min ݃(݂݊݅ = )ݔ ௫א

݂݅݊

ఌஹ ௫אௌషభ,ಾశഄ

݃( = )ݔ൬ ݉݅݊ ܿ ൰ ݂݅݊ ݄(1 + ܯ+ ߝ) = ଵஸஸ

൬ ݉݅݊ ܿ ൰ ݄(1 + )ܯ. ଵஸஸ

ఌஹ

92

Chapter Three

Observe that the minimum of ݃ on the unbounded closed subset ܣெ is attained at one of the extreme points of ܣெ . Latter points are exactly the extreme points of ܵିଵ,ெ . 7KHFRQFOXVLRQIROORZVƑ Corollary 4.2.2. Let ݊ 2 be a natural number and {ܿଵ , … , ܿ } ؿ (0, λ), 0, ݆ = 1, … , ݊, σୀଵ = 1. Then the following relation holds:

ܿ ݈݊൫1 + ൯ ൬ min ܿ ൰ ݈݊(2) ଵஸஸ

ୀଵ

where equality occurs if and only if = 0, ݆ { א1, … , ݊}\{݆ }, బ = 1, and ܿబ = ݉݅݊ ܿ for some ݆ { א1, … , ݊}. ଵஸஸ

Corollary 4.2.3. Let ݄ be a concave continuous strictly increasing function on [1, λ), (ܿ )ஹଵ a bounded sequence of positive numbers such that ݂݅݊ ܿ > 0. Assume that ஹଵ

݄(1 + ݔ < )ݔ, > ݔ0, ݄(1) = 0. Then for any convergent series σୀଵ ݔ = ܯof nonnegative numbers, we have

ܿ ݄(1 + ݔ ) ൬݂݅݊ ܿ ൰ ݄(1 + )ܯ ஹଵ

ୀଵ

Proof. Observe the series in the left-hand side of the above inequality is convergent, because of

ܿ ݄(1 + ݔ ) ൬ܿ ݑݏ ൰ ݔ = ܯ൬ܿ ݑݏ ൰ < λ. ୀଵ

ஹଵ

ୀଵ

ஹଵ

For each ݊ 2, from Theorem 4.2.1 we know that σୀଵ ܿ ݄൫1 + ݔ ൯ ൬ ݉݅݊ ܿ ൰ ݄൫1 + σୀଵ ݔ ൯. Passing through the limit over ݊ ՜ λ, one ଵஸஸ

obtains

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93

ܿ ݄(1 + ݔ ) = lim ൮ ܿ ݄൫1 + ݔ ൯൲ ՜

ୀଵ

ୀଵ

lim ൬ ݉݅݊ ܿ ൰ ݄ ቌ1 + ݔ ቍ = ՜ ଵஸஸ

ୀଵ

൬݂݅݊ ܿ ൰ ݄(1 + )ܯ. ஹଵ

7KLVFRQFOXGHVWKHSURRIƑ Theorem 4.2.4. Let ݄ be a strictly increasing strictly concave continuous real function on [1, λ), such that ݄(1) = 0, and the associated function ݑ՜ ݑή ݄(1 + 1Τ )ݑis strictly increasing on (0, λ). (a) Let ݊ אԳ, ݊ 2, {ܿଵ , … , ܿ } ( ؿ0, ), ݔ 0, ݆ { א1, … , ݊} be such that σୀଵ ܿ ݔ = ܯ, , where ܯ > 0 is a given constant. Then we have

ܿ ή ݄൫1 + ݔ ൯ ൬ ݉݅݊ ܿ ൰ ή ݄ ቌ1 + ୀଵ

ଵஸஸ

ܯ ቍ ݉݅݊ ܿ

ଵஸஸ

and equality occurs if and only if ݔ( = ݔଵ , … , ݔ ) = ൬0, … ,0,

ெ ೕబ

, 0, , ,0൰,

where ܿబ = ݉݅݊ ܿ , and the non- null component(s) of ݔis (are) ݔబ . ଵஸஸ

(b) If (ܿ )ஹଵ is a bounded sequence of positive numbers with ݂݅݊ ܿ > ஹଵ

0, assuming that ݄(1 + ݔ < )ݔ, > ݔ0, then for any sequence (ݔ )ஹଵ of nonnegative numbers such that the sum σୀଵ ܿ ݔ = ( א ܯ0, λ),we have

ܿ ݄(1 + ݔ ) ൬݂݅݊ ܿ ൰ ݄ ቌ1 + ୀଵ

ஹଵ

ܯ ቍ ݂݅݊ ܿ ஹଵ

Proof. To prove (a), one repeats the idea of the proof of Theorem 4.2.1, where we define the simplex

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Chapter Three

ܵିଵ ؔ ቊݔ( = ݔଵ , … , ݔ ) אԹ ; ݔ 0, ݆ { א1, … , ݊},

ୀଵ

Then ܵ(ݔܧିଵ ) = {݁ଵ , … , ݁ }, ݁ଵ ؔ ቀ

ெ భ

ܿ ݔ = ܯ ቋ.

, 0, … ,0ቁ , … , ݁ ؔ ቀ0, … ,0,

ெ

ቁ,

݃ ( ؔ )ݔ ܿ ή ݄൫1 + ݔ ൯, ݔ( = ݔଵ , … , ݔ ) ܵ אିଵ ֜ ݃ ൫݁ ൯ ୀଵ

= ܿ ݄ ቆ1 +

ܯ ቇ , ݆ = 1, … , ݊. ܿ

Due to concavity property of ݃, one has min ݃ ( = )ݔmin ݃ ൫݁ ൯ = min ܿ ݄ ቆ1 +

௫אௌషభ

ଵஸஸ

൬ ݉݅݊ ܿ ൰ ή ݄ ቌ1 + ଵஸஸ

ଵஸஸ

ܯ ቇ= ܿ

ܯ ቍ, ݉݅݊ ܿ

ଵஸஸ

where the last inequality follows from the hypothesis on the function ݑ՜ ݑή ݄(1 + 1Τ)ݑ, > ݑ0, which was assumed to be strictly increasing. The last assertion from (a) as well as that from (b) can be deduced in a similar way to the proof of Theorems 4.2.1 and Corollary 4.2.3, by means what we have already discussed of in the present proof. Ƒ Example 4.2.1. All assumptions in the statement of Theorem 4.2.1 are valid in the particular case when ݄()ݐ(݈݊ = )ݐ, > ݐ0. Example 4.2.2. The function ݄( ݐ = )ݐ െ 1, ݐ 0, 0 < < 1 satisfies all requirements of Theorem 4.2.1. Verifying these assertions is an elementary task. Therefore, we shall omit the proof. In particular, Theorem 4.2.1 can be applied for these functions taken as ݄. Remark 4.2.1. The inequality of Theorem 4.2.4 point (a) remains valid under the weaker constraints ݔ 0, ݆ { א1, … , ݊}, σୀଵ ܿ ݔ ܯ > 0. The next result refers to some properties of the function ߩ( ݑ ؔ )ݑή ݈݊(1 + 1Τ)ݑ, > ݑ0. The proof will be omitted because is too elementary.

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Proposition 4.2.5. (a) The function ߩ is strictly increasing, strictly concave on (0, λ), and the horizontal line of equation = ݕ1 is an asymptote for the graph of the function ߩ at infinity. ଵ

(b) The unique fixed point ݑ ( א0, λ) of the function ߩ is ݑ = . The ିଵ function ߩ is a contraction from [1Τ(݁ െ 1), λ) to [1Τ(݁ െ 1), 1) ؿ ଵ ଵ [1Τ(݁ െ 1), λ), of contraction constant ߩᇱ ቀ ቁ = . Choosing ିଵ ݑ = 1, ݑାଵ = ߩ(ݑ ), ݊ אԳ, we have the following well-known relation controlling the speed of convergence ݑ ՜ ݑ: ฬݑ െ

1 ݁ ି ฬ ൫1 െ ݈݊(2)൯, ݊ אԳ. 1 െ ݁ ିଵ ݁െ1

(c) Denote by ࣯ the convex cone of all strictly increasing functions ݑ: (0, λ) ՜ (0, λ), and also denote by ߩු: ࣯ ՜ ࣯, ߩු(ל ߩ ؔ )ݑ ݑ, ቆߩ൫)ݔ(ݑ൯ = ݈݊)ݔ(ݑቀ1 +

ଵ ௨(௫)

ቁ , ( א ݔ0, λ)ቇ. Then the unique

fixed point of ߩු is the constant function = )ݔ(ݑ

ଵ ିଵ

, > ݔ0.

(d) The unique self-adjoint operator ܣwith the spectrum ߪ(( ؿ )ܣ0, λ) which verifies the equality ܣή ݈݊( ܫ+ ିܣଵ ) = ܣ is = ܣ

ଵ ିଵ

ܫ.

(e) The convex cone of all strictly increasing, strictly concave continuous functions from (0, λ) to itself is closed with respect to the operation of composition of functions. The set of all continuous strictly increasing strictly concave functions from (0, λ) to itself, having a common fixed point, is convex and closed with respect to composition of functions. Similar results hold true for the function ߩ (((ݑ = )ݑ1 + 1/)ݑ െ 1), > ݑ0, 0 < < 1. The unique fixed point of this function is ݑ = 1Τ൫2ଵΤ െ 1൯ ( א0,1). Observe that ՝ 0 ֞ ݑ ՝ 0,

՛ 1 ֞ ݑ ՛ 1.

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Chapter Three

4.3. An operatorial variant We start this section by recalling some known results on self-adjoint (linear) operators acting on an arbitrary complex Hilbert space ܪ. Let ࣛ be the real vector space of self-adjoint operators from ܪto itself. Then ࣛ is an ordered vector space, endowed with the order relation defined by ܷ ܸ ֞ < ܷ(݄), ݄ > < ܸ(݄), ݄ >, ܪ א ݄, ܷ, ܸ ࣛ א Unfortunately, for arbitrary ܷ, ܸ ࣛ א, the supremum ܷ{ݑݏ, ܸ} = ܷ ܸ ש or/and the infimum ݂݅݊{ܷ, ܸ} = ܷ ܸ רmight not exist in ࣛ. However, the following main known result holds true. Theorem 4.3.1. Let (ܷ )ஹ be a monotone nondecreasing bounded above sequence of operators in ࣛ. Then there exists ܷ = ܷ ݑݏ in ࣛ and ܷ is ஹ

the pointwise limit of the sequence ݈݅݉ԡܷ (݄) െ ܷ(݄)ԡ = 0, ܪ א ݄. ( ܷ )ஹ :՜ஶ Obviously, a similar conclusion follows for decreasing bounded below sequences (ܷ )ஹ of elements of ࣛ. Remark 4.3.1. A converse-type result related to Theorem 4.3.1 holds true in a much more general setting and its proof is obvious. Namely, let ܻ be an ordered vector space, which is also a topological vector space such that the positive cone ܻା is topologically closed. If (ܷ )ஹ is an increasing sequence in ܻ such that there exists ܷ = lim ܷ ܻ א, then there exists ՜ஶ

ܷ ݑݏ and ܷ ݑݏ = ܷ. ஹ

ஹ

To avoid the fact that ࣛ is not a vector lattice, as well as the non commutativeness of multiplication of elements from ࣛ, for any ࣛ א ܣone uses the construction of the following space ܻ = ܻ()ܣ. Theorem 4.3.2. Let ܷ ࣛ א, ܻଵ ؔ {ܸ }ܸܷ = ܷܸ ;ࣛ א, ܻ = ܻ(ܷ) ؔ {ܹ א ܻଵ ; ܸܹ = ܹܸ, ܻ א ܸଵ }. Then ܻ is a commutative (real) Banach algebra and an order complete Banach lattice, where: |ܸ| ؔ ܸ{ݑݏ, െܸ} = ඥܸ ଶ , ܻ א ܸ

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(|ܸ| is equal to the positive square root of the positive self-adjoint operator ܸ ଶ ). The proof of Theorem 4.3.2 can be found in [17]. Having in mind these background-type results and the above notations, we can prove the next main new theorem of this chapter. In the sequel, for ࣛ א ܣ, the space ܻ = ܻ( )ܣwill be that defined in Theorem 4.3.2. For a continuous real function ݄ on the spectrum ߪ( )ܣof an operator ࣛ א ܣ, we will denote also by ݄ the mapping obtained from ݄ by means of functional calculus attached to ܣ. Theorem 4.3.3. Let ࣛ א ܣbe such that the spectrum ߪ(( ؿ )ܣ0, λ), (ܶ )ஹଵ a sequence of elements in ܻ( )ܣfor which there exist ܽ, ܾ אԹ, 0 < ܽ < ܾ, with the property that the spectrums ߪ(ܶ ) ܽ[ ؿ, ܾ], ݊ אԳ\{0}. Let ݄ be a concave continuous increasing function on [1, λ) such that 0 ݄(1 + )ݔ ݔ, ݔ 0, ݄(1) = 0, ݄( = )ܫ, and σஶ ୀଵ ݔ a convergent numerical series of positive terms. Denote ؔ ݏσஶ ୀଵ ݔ . Suppose there exists ( א ܮ0, λ) such that |݄(1 + ݑଵ )ݐെ ݄(1 + ݑଶ |)ݐ ݑ|ܮଵ െ ݑଶ |, ݑ{ଵ , ݑଶ } ؿԹା , )ܣ(ߪ א ݐ. Then the following inequality holds ஶ

ܶ ݄( ܫ+ ݔ )ܣ ൬݂݅݊ ܶ ൰ ݄( ܫ+ )ܣݏ ܽ ή ݄( ܫ+ ܻ א )ܣݏା \{0}. (3.4) ஹଵ

ୀଵ

(I: ܪ՜ ܪis the identity operator). Proof. Observe the conditions on the spectrums of ܶ im ply: ܶ =

න ்ܧ݀ݐ ܽ න ்݀ܧ = ܽܫ, א ݊Գ\{0} ֜ ݂݅݊ ܶ ܽܫ ఙ(்)

ஹଵ

ఙ(் )

ܻ אା \{0}, where ܧ is the (positive) spectral measure attached to the self-adjoint operator ܷ ܻ א. Similarly, we have ܶ ܾ ܫfor all natural numbers ݊ 1, hence ܶ ݑݏ ܾܻ א ܫା . On the other hand, one has ஹଵ

݄( ܫ+ ݔ = )ܣන ݄(1 + ݔ ܧ݀ )ݐ න ݔ ܧ݀ ݐ = ఙ()

ఙ()

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Chapter Three

ݔ න ܧ݀ݐ = ݔ ܣ, ݊ 1, ݊ אԳ ఙ()

Consequently, also using Theorem 4.3.1, we derive that the series σஶ ୀଵ ܶ ݄(1 + ݔ )ܣis pointwise convergent to an element of ܻା , since ஶ

ஶ

ܶ ݄( ܫ+ ݔ )ܣ ൬ܶ ݑݏ ൰ ൭ ݔ ൱ ܣ ܾܻ א ܣݏܾ = ܣݏܫା . ஹଵ

ୀଵ

ୀଵ

The next step is to prove a result similar to that from Theorem 4.2.1, in the operatorial setting. The conclusion of the present theorem will follow via a passing to the limit operation. Let ݊ 2 be an arbitrary natural number,

ܵିଵ ؔ ቐݕ( = ݕଵ , … , ݕ ) א

ݏ ؔ ݔ , ୀଵ

Թା ; ݕ

= ݏ ቑ,

ୀଵ

݃ ( ؔ )ݕ ܶ ݄൫ ܫ+ ݕ ܣ൯, ܵ א ݕିଵ

(3.5)

ୀଵ

Obviously, for any fixed > ݐ0, the real function ݑ՜ ݄(1 + )ݐݑis concave on Թା , so that ݕ՜ ߮(݄ = )ݕ൫1 + ݕ ݐ൯ is concave on ܵିଵ . Let {ߙଵ , ߙଶ } ؿ Թା , ߙଵ + ߙଶ = 1, ( ݕ) ܵ אିଵ , ݇ { א1,2}. The following relations hold (ଵ)

߮൫ߙଵ ( ݕଵ) + ߙଶ ( ݕଶ) ൯ ؔ ݄൫ ܫ+ ൫ߙଵ ݕ (ଵ)

න ݄൫1 + ൫ߙଵ ݕ

(ଶ)

+ ߙଶ ݕ ൯ܣ൯ =

(ଶ)

+ ߙଶ ݕ ൯ݐ൯ ݀ܧ =

ఙ()

(ଵ)

(ଶ)

න ݄ ቀߙଵ ൫1 + ݕ ݐ൯ + ߙଶ ൫1 + ݕ ݐ൯ቁ ݀ܧ ఙ()

(ଵ)

(ଶ)

ߙଵ න ݄ ൫1 + ݕ ݐ൯݀ܧ + ߙଶ න ݄ ൫1 + ݕ ݐ൯݀ܧ = ఙ()

ఙ()

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99

(ଶ)

ߙଵ ݄൫ ܫ+ ݕ ܣ൯ + ߙଶ ݄൫ ܫ+ ݕ ܣ൯ = ߙଵ ߮൫( ݕଵ) ൯ + ߙଶ ߮൫( ݕଶ) ൯ Hence ߮: ܵିଵ ՜ ܻ is concave. It follows that ߰: ܵିଵ ՜ ܻ, ߰(ؔ )ݕ ܶ߮ ( )ݕis also concave for any ܶ ܻ אା (one can multiply by a positive operator ܶ both members of an inequality, because the product of two selfadjoint positive permutable operators is self-adjoint and positive). On the other hand, it is straightforward that a finite sum of concave operators from a convex subset of a vector space to an ordered vector space is concave. It results that the operator ݃ : ܵିଵ ՜ ܻା defined by (3.5) is concave. On the other hand, the set of extreme points of ܵିଵ is ܵ(ݔܧିଵ ) = {݁ଵ , … , ݁ }, where ݁ଵ = (ݏ , 0, … ,0), … , ݁ = (0, … ,0, ݏ ) Let ܵ א ݔିଵ . Carathéodory’s theorem leads to the existence of {ߙଵ , … , ߙ } ؿԹା , σୀଵ ߙ = 1, such that = ݔσୀଵ ߙ ݁ . Now we apply Jensen inequality for the concave operator ݃ (which can be proved by induction, as in the case of concave real functions). Obviously, one has ݄( = )ܫ ֜ ݃ ൫݁ ൯ = ܶ ݄( ܫ+ ݏ )ܣ, ݆ { א1, … , ݊}. It results

݃ ( )ݔ ߙ ݃ ൫݁ ൯ = ߙ ܶ ݄( ܫ+ ݏ )ܣ ൬ inf ܶ ൰ ݄( ܫ+ ݏ )ܣ ୀଵ

ୀଵ

ଵஸஸ

To conclude the proof, observe that the Lipchitz condition on ݕ՜ ݄(1 + )ݐݕin variable ݕ, uniformly with respect to parameter )ܣ(ߪ א ݐleads to: ݏ ՜ ֜ ݏԡ݄( ܫ+ ݏ )ܣെ ݄( ܫ+ )ܣݏԡ = (݄| ݑݏ1 + ݏ )ݐെ ݄(1 + |)ݐݏ ݏ|ܮ െ |ݏ՜ 0, ݊ ՜ λ

௧אఙ()

This proves that ݏ ՜ ֜ ݏlim ݄( ܫ+ ݏ ܫ(݄ = )ܣ+ )ܣݏ ՜ஶ

Passing to the limit, also using (3.6), one obtains

(3.6)

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Chapter Three

ஶ

ܶ ݄( ܫ+ ݔ ݈݉݅ = )ܣ ܶ ݄൫ ܫ+ ݔ ܣ൯ = ݈݅݉ ݃ ( )ݔ

ୀଵ

ୀଵ

lim ൬ inf ܶ ൰ ݄( ܫ+ ݏ )ܣ ൬݂݅݊ ܶ ൰ ቆ݈݄݅݉( ܫ+ ݏ )ܣቇ =

ଵஸஸ

ஹଵ

൬݂݅݊ ܶ ൰ ݄( ܫ+ )ܣݏ ஹଵ

where all the limits are considered in the topology of pointwise convergence. The last relation in (2) follows too, because of ݂݅݊ ܶ ܽܫ. ஹଵ

Ƒ

Remark 4.3.1. With the notations and under hypothesis of Theorem 4.3.3, consider the function ݑ՜ ݈݊(1 + )ݐݑ, א ݑԹା , ( ؿ )ܣ(ߪ א ݐ0, λ). Application of Lagrange theorem yields: ݐ |ݑଵ െ ݑଶ | ఏவ ௧אఙ() 1 + ߠݐ

|݈݊(1 + ݑଵ )ݐെ ݈݊(1 + ݑଶ |)ݐ ݑݏmax ԡܣԡ|ݑଵ െ ݑଶ |

Similarly, for 0 < < 1, considering the function ݑ՜ ݄(1 + ؔ )ݐݑ (1 + )ݐݑ െ 1, א ݑԹା , ( ؿ )ܣ(ߪ א ݐ0, λ), one obtains |݄(1 + ݑଵ )ݐെ ݄(1 + ݑଶ (| = |)ݐ1 + ݑଵ )ݐ െ (1 + ݑଶ )ݐ | = ฬ

ݑ(ݐଵ െ ݑଶ )ฬ ԡܣԡ|ݑଵ െ ݑଶ | = ݑ|ܮଵ െ ݑଶ |, (1 + ߠ)ݐଵି

where the Lipchitz constant ؔ ܮԡܣԡ related to the variable ݑdoes not depend on the parameter )ܣ(ߪ א ݐ. Thus both examples 4.2.1 and 4.2.2 for ݄ are suitable for the operatorial version of the first inequality (3.4) proved in Theorem 4.3.3. Corollary 4.3.4. Under the hypothesis and using the notations from Theorem 4.3.3, the following relations hold true ஶ

ܶ ݈݊( ܫ+ ݔ )ܣ ܽ ܫ(ܣݏ+ ି)ܣݏଵ ୀଵ

ܽ߱ݏ ܫ 1 + ݏԡܣԡ

Earlier and Recent Results on Convexity and Optimization

101

where ߱ ؔ ݂݅݊ ߪ( > )ܣ0. Proof. Observe that for any )ܣ(ߪ א ݐone has ݈݊(1 + (ݐݏ > )ݐݏ1 + ି)ݐݏଵ ,

ݐݏ ݏή sup ߪ(ݏ = )ܣԡܣԡ

We have ߪ( ܫ+ = )ܣݏ1 + [ ؿ )ܣ(ߪݏ1 + ߱ݏ, 1 + ݏԡܣԡ]. In particular, 0 is not an element of ߪ( ܫ+ )ܣݏ, so that ܫ+ ܣݏis invertible. Moreover, the following relations hold ߪ(( ܫ+ ି)ܣݏଵ ) = ൫ߪ( ܫ+ )ܣݏ൯

ିଵ

1 1 ؿ , ൨ 1 + ݏԡܣԡ 1 + ߱ݏ

Integrating with respect to the spectral measure ܧ one obtains ݈݊( ܫ+ = )ܣݏන ݈݊(1 + ܧ݀)ݐݏ න (ݐݏ1 + ି)ݐݏଵ ݀ܧ ఙ()

= ܫ(ܣݏ+ ି)ܣݏଵ

ఙ()

Thus, the first inequality in the statement follows from Theorem 4.3.3. For the second one, observe that (ݐݏ1 + ି)ݐݏଵ (߱ݏ1 + ݏԡܣԡ)ିଵ , ܫ(ܣݏ ֜ )ܣ(ߪ א ݐ+ ି)ܣݏଵ ߱ݏ ܫ 1 + ݏԡܣԡ 7KLVFRQFOXGHVWKHSURRIƑ

CHAPTER FOUR INVARIANT SUBSPACES AND INVARIANT BALLS OF BOUNDED LINEAR OPERATORS

1. Introduction We follow our results published in [50]. For the history of the invariant subspace problem and related results see the references in [50]. The first aim of the present work is to give solutions to the invariant subspace problem in a special Fréchet space of entire operator valued mappings. Also, one makes the connection to the hyperinvariant subspaces of an arbitrary bounded operator, by means of an example. Finally, the invariance of the unit ball of ܮଵ spaces with respect to linear bounded operators is discussed. The paper is organized as follows. Section 2 is devoted to the existence of common invariant subspaces in a Fréchet space of entire mappings. The connection with the derivation operation is briefly pointed out in Section 3. In Section 4, we discuss a related differential equation involving normal operators. Also, we give an example of a hyperinvariant subspace concerning the whole first part of this chapter. Finally, in Section 5, the invariance of the unit ball in ܮଵ spaces is discussed. This is the second aim of this chapter. To do this, we apply polynomial approximation results. Some of the results in Section 5 (such as Corollaries 5.3, 5.4, Theorem 5.6) are realized in terms of quadratic forms (with vector – coefficients).

2. Existence of invariant subspaces for special operators If ܺଵ , ܺଶ are two topological vector spaces (TVS), we denote by ܺ(ܥܮଵ , ܺଶ ) the space of all linear continuous operators applying ܺଵ into ܺଶ . If ܺଵ = ܺଶ , one denotes ܺ(ܥܮଵ ) ؔ ܺ(ܥܮଵ , ܺଵ ). If ि ܺ(ܥܮ كଵ ) is a nonempty subset, we denote िଶ = ൛ܷଵ ܷଶ ; ܷ אि, ݆ = 1,2ൟ. Proposition 2.1. Let ܺଵ be a TVS, such that for any ि ܺ(ܥܮ كଵ ), ि ଶ كि, there exists a closed proper common invariant subspace for all ܷ אि. Then any topological vector space ܺ linearly and topologically isomorphic to ܺଵ has the corresponding similar property.

Invariant Subspaces and Invariant Balls of Bounded Linear Operators

103

Proof. Let ܺଵ , ܺ be as in the statement of the proposition 2.1. Consider a linear topological isomorphism ߮ ܺ(݉ݏܫ א, ܺଵ ). Let ܶ אॎ )ܺ(ܥܮ ك. Then the mapping Ȱ: )ܺ(ܥܮ՜ ܺ(ܥܮଵ ) Ȱ(ܶ) = ߮ ି ߮ ל ܶ לଵ = ܷ verifies Ȱ(ܶଵ ܶଶ ) = Ȱ(ܶଵ )Ȱ(ܶଶ ), ܶ אॎ, ݆ = 1,2, that is Ȱ is multiplicative, and so is Ȱିଵ . It follows that Ȱ applies any subset ॎ )ܺ(ܥܮ ك, ॎଶ كॎ, onto a corresponding subset ि = Ȱ(ॎ) ك ܺ(ܥܮଵ ), and the corresponding assertion on Ȱିଵ holds. Moreover, the following relations are true ଶ

िଶ = ൫Ȱ(ॎ)൯ = Ȱ(ॎ ଶ ) كȰ(ॎ) = ि. On the other hand, by hypothesis, there exists a closed proper common invariant subspace ܵ ܺ ؿଵ of all operators ܷ = Ȱ(ܶ) אȰ(ॎ). Then ܵሚ = ߮ ିଵ (ܵ) is a closed proper invariant subspace of all operators ܶ אॎ. ᇝ Let ܨbe a complex Banach space and )ܨ(ܤthe space of all linear bounded operators acting on ܨ. As a model of a concrete Fréchet space, we consider the space ܺଵ = ܧ = ܧ൫ԧ, )ܨ(ܤ൯ of all entire functions, having the form ஶ

݂: ԧ ՜ )ܨ(ܤ, ݂( = )ݖ ܷ ݖ ,

݈݅݉ ݑݏඥԡܷ ԡ = 0

(4.1)

ୀ

The space ܧ, endowed with the sequence of seminorms ݍ (݂) = sup ԡ݂()ݖԡ , ݂ ܧ א, ݊ אԳ, ݊ 1, ݂ ܧ א |௭|ஸ

is a Fréchet space. This is a consequence of the Cauchy formula, which works for the elements of ܧ. The topology defined by these seminorms is the topology of uniform convergence on compact subsets of ԧ.

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Lemma 2.2. Let ݂ ܧ א, א ݖԧ. Then for any admissible contour ߛ surrounding ݖ, Cauchy’s formula is true:

݂(= )ݖ

݂(ߞ) 1 න ݀ߞ 2ߨ݅ ߞ െ ݖ ఊ

Proof. If ݂ is given by (4.1), due to the uniform convergence of the series from below on any compact subset of ԧ , the following equalities hold ஶ

ஶ

݂( = )ݖ ܷ = ݖ ܷ ୀ

ୀ

=

σஶ 1 ߞ 1 ୀ ܷ ߞ න ݀ߞ = න ݀ߞ ߞെݖ 2ߨ݅ ߞ െ ݖ 2ߨ݅ ఊ

ఊ

1 ݂(ߞ) න ݀ߞ. 2ߨ݅ ߞ െ ݖ ఊ

All the series appearing above are absolutely convergent with respect to the operatorial norm on )ܨ(ܤ. 7KLVFRQFOXGHVWKHSURRIƑ Now the fact that ܧis complete can be proved as in the case of complex valued functions, using Cauchy’s criterion for sequences of applications with values in Banach spaces and the implication ݂ ՜ ݂ ֜ ݂ᇱ ՜ ݂ ᇱ in ܧ. This last implication is a consequence of the Cauchy formula. Also notice that the preceding result leads to the derivation term by term of a function ݂ ܧ א, where ܧis given by means of (1). If ܷ )ܨ(ܤ א, then the operator ܩ defined by ܩ (݂)()ݖ(݂ܷ = )ݖ, א ݖԧ, ݂ ܧ א. is an element of )ܧ(ܥܮ. Namely, the following evaluation holds true ݍ ൫ܩ (݂)൯ ԡܷԡݍ (݂), ݂ ܧ א, ݊ אԳ, ݊ 1. Theorem 2.3. Let ॏଶ كॏ )ܨ(ܤ ك. Then there exists closed proper common invariant subspaces in ܧ, of all operators ܩ , ܷ אॏ. Proof. Define ܵ ܧ ؿas the vector subspace generated by all applications having the form

Invariant Subspaces and Invariant Balls of Bounded Linear Operators

105

ܷ݂()ܮݖ, ܷ אॏ, ݂ ܧ א , े א ܮ, א ݖԧ, where ܧ is a nonempty subset of entire complex functions vanishing at the ෩ be an arbitrary origin, े is an arbitrary nonempty subset of )ܨ(ܤ. Now let ܷ element of ॏ, and ݃ ܵ א , ݃( = )ݖ ܿ ܷ ݂ (ܮݖ ), ܿ אԧ, ܷ אॏ, ݂ ܧ א , ܮ े א, א ݖԧ.

Then ෩ܷ ݂ (ܮݖ ), א ݖԧ, ܩ෩ (݃)( = )ݖ ܿ ܷ

෩ܷ אॏ , so that one deduces that ܩ෩ ( ܵ ) ܵ ك . where the sum is finite, ܷ Since we have already remarked that ܩ෩ is continuous on ܧ, we infer that ܩ෩ (ܵҧ ) ܵ كҧ . It remains to prove that ܵҧ ് ܧ, as we now show. Notice that the convergence in ܧimplies the pointwise convergence, so that any element of ܵҧ is vanishing at the origin, as being the limit of a sequence of functions having this property. On the other hand, the hyperplane ܪ of all functions in ܧvanishing at the origin is closed in ܧ, by the same reason. The conclusion is that the common closed subspace ܵҧ , which is invariant ෩ אॏ, is contained in the closed hyperplane ܪ . for all the elements ܩ෩ , ܷ Hence, it is a closed proper common invariant subspace. This concludes the SURRIƑ Corollary 2.4. Let ܷ )ܨ(ܤ א. Then there exist common closed proper invariant subspaces of all operators ܩ , where ܷ )ܨ(ܤ אis such that ܷ commutes with ܷ . Proof. One applies Theorem 2.1 to the algebra ॏ of all operators commuting with ܷ . Obviously, the equality ॏଶ = ॏ KROGVƑ Corollary 2.5. There exist common proper closed invariant subspaces of all operators ܩ , when ܷ )ܨ(ܤ א. Proof. One applies Theorem 2.1 to ॏ = )ܨ(ܤ. Ƒ

106

Chapter Four

3. Connections with classical operations Let ܷ )ܨ(ܤ אbe an invertible operator, ܵ = {݂ = )ݖ(݂ ;ܧ א ܷ ݁)ܷݖ(ݔ, א ݖԧ, ݇ אԺ}, ܵሚ = ܿ )ܵ(be the convex hull of ܵ. Let א ܦ )ܧ(ܥܮbe the derivation operator and ି ܷ = )ݖ()݃(ܬଵ ݃()ݖ, א ݖԧ, ݃ ܵ אሚ. ܬ stands for an integration operation. Remark 3.1. The following assertion holds true: ݃(ܦҧ ) = ܩ (݃ҧ ), ݃(ܬҧ ) = ܩ షభ (݃ҧ ) for all applications ݃ҧ in the topological closure of ܵሚ. In fact, a straightforward calculation shows that ܩ = ܦ on ܵሚ. Since both operators ܦ, ܩ are continuous on ܵሚ, their equality on the topological closure of ܵሚ follows. Obviously, ିܬ = ܦଵ on ܵሚ, and the equality can be extended to the closure of ܵሚ. The last equality in the statement is obvious.

4. A related differential equation In the next result we give a new result involving normal operators. Let ܪ be an arbitrary Hilbert space and ܶ )ܪ(ܤ אa normal operator. Consider the differential equation ܶ ܻ () + ܶ ିଵ ܻ (ିଵ) + ڮ+ ܻܶ ᇱ െ ܻ = 0

(4.2)

where ܻ: ԧ ՜ )ܪ(ܤis the unknown mapping? Theorem 4.1. The following statements are equivalent (a) ܻ() څ ܶݖ߱(ݔ݁ = )ݖ, ߱ > 0 is a solution of the differential equation (4.2); (b) there exists a unitary operator ܷ )ܪ(ܤ אsuch that ܶ = (߱ିଵ ݐ )ଵ/ଶ ܷ, where ݐ is the unique positive root of the algebraic equation ܲ( ݐ ؔ )ݐ + ݐିଵ + ڮ+ ݐെ 1 = 0

(4.3)

Invariant Subspaces and Invariant Balls of Bounded Linear Operators

107

Proof. In order to prove (a)֜(b), we compute the left-hand side member of (4.2), using the hypothesis. One deduces (ܶ ߱ ܶ څ + ڮ+ ܶ߱ܶ څെ = ) څ ܶݖ߱(ݔ݁)ܫ0. Multiplying with ݁(ݔെ߱ ) څ ܶݖand using the fact that ܶ is normal, the last equality can be rewritten as (߱ܶܶ ) څ + ڮ+ (߱ܶܶ ܫ = ) څ. Let ܣଵ = (ܶܶ ) څଵ/ଶ be the positive square root of the self - adjoint positive operator ܶܶ څ. The last relation yields ܲଵ (߱ܣଵଶ ) = (߱ܣଵଶ ) + ڮ+ (߱ܣଵଶ ) = ܫ. This equality further yield ܲଵ ൫ߪ(߱ܣଵଶ )൯ = ߪ൫ܲଵ (߱ܣଵଶ )൯ = ߪ( = )ܫ1. Since all the operators involved in the last equality are self - adjoint, one deduces that ܲ(ܲ = )ݐଵ ( )ݐെ 1 = 0, ܣ߱(ߪ א ݐଵଶ ), where ܲ is defined by (4.3). Since ߪ(߱ܣଵଶ ) consists only in positive numbers, and the equation ܲ( = )ݐ0 has a single positive root ݐ we infer that ߪ(߱ܣଵଶ ) = {ݐ }, where ݐ is the positive root of equation (4.3). The last equality shows that ߱ܣଵଶ = ݐ ܫ. Following the preceding notations, this equality can be rewritten as (߱ݐିଵ )ଵ/ଶ ܶ(߱ݐିଵ )ଵ/ଶ ܶ ܫ = څ,

108

Chapter Four

or, equivalently, ܷܷ ܫ = ܷ څ ܷ = څ, where ܷ = (߱ݐିଵ )ଵ/ଶ ܶ, ܶ = څ (߱ିଵ ݐ )ଵ/ଶ ܷ څ. Thus (a)֜(b) is proved. Conversely, assume that (b) holds. We have ܶ ܻ () + ܶ ିଵ ܻ (ିଵ) + ڮ+ ܻܶ ᇱ െ ܻ =

ቀ(߱ିଵ ݐ ) ଶ ܷ ߱ (߱ିଵ ݐ ) ଶ ܷ څ + ڮ+ (߱ିଵ ݐ )ܷܷ߱ څ ଵ

െ ܫቁ ݁ ݔ൬(߱ݐ )ଶ څ ܷݖ൰ = భ

= ܲ(ݐ )݁ ݔቀ(߱ݐ )మ څ ܷݖቁ = 0. Ƒ Example 4.1. The vector space ܵ of all the solutions of (4.2) is invariant with respect to the derivation operation and hyperinvariant with respect to ்ܩ. This assertion holds true since the derivation - operator commutes with any operator not depending on ݖ. Consider the space ܧof all entire functions from the complex plane into )ܪ(ܤ, endowed with the topology of uniform convergence on the compact subsets of ԧ. This is a Fréchet space, on which the derivation operation is continuous (from ܧonto )ܧ. Hence, the kernel of the linear differential operator involved in (4.2) is a closed subspace of ܧ, so that the subspace ܵ of all the solutions of (4.2) is closed. In order to formalize the above assertion, for ܷ )ܪ(ܤ אlet ܩ ܧ ՜ ܧbe the linear continuous operator defined by G (݂)()ݖ(݂ܷ = )ݖ, א ݖԧ. If ܷ commutes with ܶ and ܻ ܵ א, one deduces ()

ܶ ൫G (ܻ)൯

ᇱ

+ ڮ+ ܶ൫G (ܻ)൯ െ G (ܻ) =

= ܶ (ܷܻ)() + ڮ+ ܶ(ܷܻ)ᇱ െ ܷܻ = ܷ൫ܶ ܻ () + ڮ+ ܻܶ ᇱ െ ܻ൯ = 0. Hence ܩ (ܻ) ܵ אfor all ܻ ܵ אand all operators ܷ )ܪ(ܤ אcommuting with ܶ. Notice that this example does not depend on the proof of Theorem 2.3. It works for an arbitrary, not necessarily normal operator ܶ )ܪ(ܤ א. Remark 4.1. If ݊ = 2, we have ݐଶ = ൫െ1 + ξ5൯/2.

5. On the invariance of the unit ball in spaces Using polynomial approximation on unbounded subsets reviewed in Chapter 1, Section 4, one can prove the following results.

Invariant Subspaces and Invariant Balls of Bounded Linear Operators

109

Let ܻ be an order complete Banach lattice. Let ܺ = ܮଵఔ ()ܣ, where ܣis a closed unbounded subset of Թ , ߥ is a ܯെ determinate positive regular Borel measure on ܣ, with finite absolute moments of all orders. As usual, ࣪ is the subspace of polynomials on ܣ. Define the set ܵଵା = {ܶ ܺ(ܤ א, ܻ); ܶ(߮) 0 ܺ א ߮ା , ԡܶԡ 1}. Theorem 5.1. For a linear operator ܸ: ࣪ ՜ ܻ, the following statements are equivalent (a) ܸ has a linear positive extension ܸ෨ ܵ אଵା ; (b) there exists ܶ ܵ אଵା such that 0 ܸ( ) ܶ(),

࣪ א ା .

We proved Theorem 5.1 in [48]. Denote ܵଵା (ܺ) ؔ ൛ܶ ܤ אା (ܺ, ܺ); ܶ൫ܤതଵ, ൯ ܤ كതଵ, ൟ, where ܺ is as above, and ܤതଵ, is the closed unit ball in ܺ. Let ܸ: ܲ ՜ ܺ be a linear operator. Corollary 5.2. The following statements are equivalent (a) ܸ has a linear positive extension ܸ෨ ܵ אଵା (ܺ); (b) there exists ܶ ܵ אଵା (ܺ) such that 0 ܸ( ) ܶ(),

࣪ א ା .

Proof. Put in Theorem 5.1 ܻ = ܺ = ܮଵఔ ()ܣ. Then ܺ is an order complete vector lattice (in which any order convergent sequence is convergent in the norm topology cf. [69]). To prove (b)֜(a), one applies the corresponding implication from Theorem 5.1. Also observe that ฮ ܸ෨ ฮ 1 if and only if ܸ෨ ൫ܤതଵ, ൯ ܤ كതଵ, . 7KLVFRQFOXGHVWKHSURRIƑ Our next goal is to give some characterizations in terms of quadratic forms (when this fact is allowed by the form of positive polynomials by means of sums of squares).

110

Chapter Four

Corollary 5.3. Let ܺ = ܮଵఔ (Թ), where ߥ is a positive regular ܯെ determinate measure on Թ (with finite moments of all orders), ݔ (= )ݐ ݐ , א ݐԹ , ݆ אԳ. Let ܸ: ࣪ ՜ ܺ be a linear operator. The following statements are equivalent (a) ܸ has a linear positive extension ܸ෨ ܵ אଵା (ܺ); (b) there exists ܶ ܵ אଵା (ܺ) such that for any finite subset ൛ߣ ൟא ؿԹ, బ

the following relations hold ߣ ߣ ܸ൫ݔା ൯ ߣ ߣ ܶ൫ݔା ൯ ,אబ

,אబ

Proof. One applies Corollary 5.1 to ܺ = ܮଵఔ (Թ), when in Theorem 5.1 one takes ݊ = 1, = ܣԹ, also using the form of positive polynomials on the real line, as being sums of squares of some polynomials with real coefficients [1@Ƒ Corollary 5.4. Let ܺ = ܮଵఔ ([0, λ)), ߥ being a positive regular Mdeterminate measure on Թା , with finite moments of all orders. Let ݔ (= )ݐ ݐ , א ݐԹା , ݆ אԳ. Let ܸ: ࣪ ՜ ܺ be a linear operator. The following statements are equivalent (a) ܸ has a linear positive extension ܸ෨ ܵ אଵା (ܺ); (b) there exists ܶ ܵ אଵା (ܺ) such that for any finite subset ൛ߣ ൟא ؿԹ, బ

the following relations hold ߣ ߣ ܸ൫ݔାା ൯ ߣ ߣ ܶ൫ݔାା ൯, ݈ { א0,1}. ,אబ

,אబ

Proof. The proof is similar to that of Corollary 5.2., also using the form of positive polynomials on Թା [1]: )ݐ( 0 א ݐԹା if and only if there exist ଵ , ଶ אԹ[ ]ݐsuch that = )ݐ(ଵଶ ( )ݐ+ ݐଶଶ ()ݐ, ݐ 0. Ƒ As it is well known, in several dimensions, there are positive polynomials which are not sums of squares. However, using approximation results, the connection with tensor products of positive polynomials in each separate

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111

variable holds true (see Chapter 1, Section 4). Namely, one can prove the following approximation lemma. Lemma 5.5. Let ߥ = ߥଵ × ߥ × ڮ be a product of ݊ 2 ܯെ determinate positive regular Borel measures on Թ, with finite absolute moments of all natural orders. Then we can approximate any nonnegative continuous compactly supported function in ܮଵఔ (Թ ) by means of dominating sums of tensor products ଵ ٔ ٔ ڮ positive polynomial on the real line, in variable ݐ , ݆ = 1, … , ݊. A similar result to Lemma 5.5 has been stated and applied in Chapter 1, Section 4, where Թ was replaced by Թା . Let ܺ = ܮଵఔ (Թ ), ݊ 2, ߥ be as in Lemma 5.1.

ݔ (ݐଵ , … , ݐ ) = ݐଵభ ݐ ڮ , ݆ = (݆ଵ , … , ݆ ) אԳ , (ݐଵ , … , ݐ ) אԹ . Let ܸ: ࣪ ՜ ܺ be a linear operator, where ࣪ is the subspace of polynomials in ݊ real variables, with real coefficients. Theorem 5.6. The following statements are equivalent (a) ܸ has a linear positive extension ܸ෨ ܵ אଵା (ܺ); (b) 0 ܸ(࣪ א )ା and there exists ܶ ܵ אଵା (ܺ) such that for any finite subsets ܬ ؿԳ, ݇ = 1, … , ݊, and any ൛ߣೖ ൟ

ೖ אೖ

ؿԹ, ݇ = 1, … , ݊,

we have

൮ ڮ൮ ߣభ ߣభ ߣ ڮ ߣ ܸ൫ݔభ ାభ , … , ା ൯൲ ڮ൲ భ ,భ אభ

.א

൮ ڮ൮ ߣభ ߣభ ߣ ڮ ߣ ܶ൫ݔభ ାభ , … , ା ൯൲ ڮ൲ భ ,భ אభ

. א

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Chapter Four

Proof. Observe that (b) says that ܸ is nonnegative on the convex cone of nonnegative polynomials, and is dominated by ܶ on the convex cone generated by the products ଵ ٔ ٔ ڮ where each is a square of a polynomial of variable ݐ אԹ, ݇ = 1, … , ݊. In particular, such tensor products are nonnegative on Թ . Hence (a)֜(b) is ෩ In order to prove the converse, extend ܸ to a linear obvious, putting ܶ = ܸ. positive operator ܸത defined on the subspace ܺଵ of all functions from ܺ whose absolute value are dominated by a polynomial (depending on that function). Then the subspace ܺଵ contains the polynomials and the compactly supported continuous functions too. Next, we prove that ܸത (߰) ܶ(߰) for all continuous nonnegative compactly supported functions ߰. By reduction to absurd, assume the contrary. Since ܶ(߰) െ ܸത (߰) ܺ בା , a Hahn – Banach separation theorem shows that there exists a positive (continuous) linear form څ ݔon ܺ such that څ ݔ൫ܶ(߰) െ ܸത (߰)൯ < 0, that is څ ݔ൫ܶ(߰)൯ < څ ݔ൫ܸത (߰)൯

(4.4)

On the other side, by Lemma 5.5, there exists a sequence ()

,ଵ, ٔ ٔ ڮ,, ՜ ߰, ݉ ՜ λ ୀ

in the norm topology of the space ܺ, where each polynomial ,, is nonnegative on the entire real axes, ݇ = 1, … , ݊. Now one applies Fatou’s lemma for the positive functional ܸ ל څ ݔത , which can be represented by means of a positive measure. Using the second relation of the point (b), one deduces ()

ݔ൫ܸത (߰)൯ ݈݂݅݉݅݊ څ

(څ ݔ

ܸ לത ) ቌ ,ଵ, ٔ ٔ ڮ,, ቍ ୀ

()

݈݂݅݉݅݊ ( )ܶ ל څ ݔቌ ,ଵ, ٔ ٔ ڮ,, ቍ = څ ݔ൫ܶ(߰)൯, ୀ

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߰ א൫ܥ (Թ )൯ା , which contradicts (4.4). We recall that ܥ (Թ ) is the vector space of all real valued continuous compactly supported functions defined on Թ . The conclusion is that ܸത (߰) ܶ(߰) for all nonnegative continuous compactly supported function ߰. Next one shows that there exists a linear extension ෩ܸ of ܸത , from ܺ into ܺ, of norm dominated by ԡܶԡ 1. The positivity of ܸ෨ IROORZVYLDGHQVLW\DUJXPHQWV7KLVFRQFOXGHVWKHSURRIƑ

CHAPTER FIVE FROM LINEAR TO SUBLINEAR AND TO CONVEX OPERATORS

1. Introduction One of the aims of this chapter is to prove a uniformly boundedness theorem for convex continuous operators. Such results were considered in [36] and [79]. Recently, we improved such type results in [58]. There are some differences between the methods and results proved in [36], [79] and our methods/results. For example, we obtain the convexity of the neighborhood ܹ of Theorem 2.1 proved below, and its properties, without assuming that the domain space ܧis locally convex, unlike the corresponding result from [36]. Moreover, we reduce the problem to the case of symmetric convex continuous operators. The special case of uniformly bounded families of sublinear operators is discussed in detail. The second goal of this paper is to extend some inequalities involving continuous linear and sublinear operator from a small set of test points to the positive cone of the domain space. The relationship between convex and linear continuous operators is emphasized as well. The rest of the chapter is organized as follows. In Section 2, a uniformly boundedness property for convex operators is proved. Some corollaries are deduced, most of them involving sublinear operators. Section 3 is devoted to extending inequalities which hold at all extreme points of a weakly compact convex subset contained in the positive cone to all points of the positive cone. One applies Krein-Milman theorem to prove Propositions 3.2 and 3.3 of Section 3.

2. Uniformly boundedness of some families of convex operators In the sequel, ܺ will be a (not necessarily locally convex) topological vector space which cannot be expressible as the countable union of closed subsets having empty interiors, and ܻ will be a locally convex vector lattice (on which the lattice operations are continuous and there exists a fundamental

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system ࣰ of neighborhoods ܸ of 0 which are convex, closed and solid subsets, i. e. |ݕଵ | |ݕଶ |, ݕଶ ݕ ֜ ܸ אଵ )ܸ א Both spaces ܺ, ܻ are vector spaces over the real field. Consider a class ࣝ of convex continuous operators ܲ: ܺ ՜ ܻ, ܲ() = . Recall that we can always reduce the problem of proving the equicontinuity of a family of convex operators at a point ݔ ܺ אto the equicontinuity of a corresponding family of convex operators at , where each element ܲ of the latter family satisfies the condition ܲ() = (cf. [36], the proof of Theorem 3.1). It is possible that some assertions of Theorem 2.1 from below be true under more general conditions. On the other hand, the proof of this theorem is not the same as that of Theorem 3.1 [36] (see the first comments in the Introduction and compare the two statements and their proofs). Theorem 2.1. Additionally assume that for each ܸ ࣰ א, and any ܺ א ݔ, there exists a small enough positive number ݎsuch that ࣪ א ܲ ܸ א )ݔ(ܲݎ. Then for any ܸ ࣰ א, there exists a closed convex neighborhood ܹ of such that ራ ܲ(ܹ ) ܸ ؿ ࣝא

One writes ݈݅݉ ܲ( = )ݔ uniformly in ܲ ࣝ א. ௫՜

Proof. For any ܸ ࣰ אand any ܲ ࣝ א, define ܲଵ : ܺ ՜ ܻ, ܲଵ (ؔ )ݔ )ݔ(ܲ{ݑݏ, ܲ(െ})ݔ, ܺ א ݔ. The operator ܲଵ is obviously convex. An additional property of ܲଵ is ܲଵ (ܲ = )ݔଵ (െ)ݔ, ܺ א ݔ. Consequently, the ଵ ଵ codomain of ܲଵ is ܻା , since 0 = ܲଵ (0 ) = ܲଵ ቀ ݔ+ (െ)ݔቁ ଵ

ଶ

ଶ

2ܲଵ (ܲ = )ݔଵ ()ݔ, ܺ א ݔ. The operator ܲଵ is also continuous, as the least ଶ upper bound of two continuous operators, thanks to the continuity of “sup” operation from ܻ × ܻ to ܻ. The subset ܲଵିଵ (ܸ ) is closed, due to the continuity of ܲଵ . Now we prove that it is also convex. Indeed, for ݔଵ , ݔଶ א ܲଵିଵ (ܸ ), [ א ݐ0,1],. the following relations hold ܲଵ ൫(1 െ ݔ)ݐଵ + ݔݐଶ ൯ (1 െ ܲ)ݐଵ (ݔଵ ) + ܲݐଵ (ݔଶ ) ܸ א ,

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Chapter Five

since ܸ is convex and ܲଵ is convex too. Now using assumption on ܸ of being solid, it results ܲଵ ൫(1 െ ݔ)ݐଵ + ݔݐଶ ൯ ܸ א (֞ ൫(1 െ ݔ)ݐଵ + ݔݐଶ ൯ א ܲଵିଵ (ܸ )). Define ܹ ؔ ځܲ ࣝאଵିଵ (ܸ ) The subset ܹ is closed and convex, as an intersection of such subsets. Clearly ڂܲ ࣝאଵ ( ܹ ) ܸ ؿ . For any ܹ א ݔ and any ܲ ࣪ א, it results |ܲ( |)ݔ )ݔ(ܲ{ݑݏ, ܲ(െܲ = })ݔଵ (ܸ א )ݔ , ଵ

because of െܲ( )ݔ ܲ(െ)ݔ, ܺ א ݔ. Indeed, 0 = ܲ(0 ) ൫ܲ( )ݔ+ ଶ

ܲ(െ)ݔ൯, ܧ א ݔ. Having in mind the property of ܸ , we infer that ܲ(א )ݔ ܸ , ܹ א ݔ , ࣝ א ܲ. The first conclusion is ڂܹ (ܲ ࣝא ) ܸ ؿ . To finish the proof, we have to show that ܹ is a neighborhood of 0 . For any ܺ א ݔ and for any ܸ ࣰ א, there exists a sufficiently small ݎ > 0 such that ߙܲଵ (ܸ א )ݔ א ߙԹ, |ߙ| ݎ , ࣝ א ܲ. We can suppose that ݎ 1. From the preceding considerations, it results ߙ [ א0, ݎ ] [ ؿ0,1] ֜ ܲଵ (ߙܲ = )ݔଵ ൫(1 െ ߙ)0ா + ߙݔ൯ ߙܲଵ (ܸ א )ݔ ֜ ܲଵ (ߙܸ א )ݔ ߙ [ אെݎ , 0] ֜ ܲଵ (ߙܲ = )ݔଵ ൫(െߙ)(െ)ݔ൯ (െߙ)ܲଵ (െ)ݔ ݎ ܲଵ (ܸ א )ݔ , ܲ ࣝ א. ଵ

These relations lead to: ܺ א ݔ, |ߙ| ݎ ֜ ߙܹ א ݔ ֜ | א ݔఈ| ܹ ܹ݊ ؿ for sufficiently large ݊ אԳ. Consequently, the following basic relation holds true ܺ = ڂאԳ ݊ ܹ . Now recall that ܹ is closed, convex, and our assumption on ܺ yields ܹ݅݊(ݐ ) ് , so that there exists ݔ ܹ(ݐ݊݅ א ) ֜ ଵ 0ா = ൫ݔ + (െݔ )൯ ܹ(ݐ݊݅ א ). This concludeVWKHSURRIƑ ଶ

Corollary 2.2. Let ܺ be a Banach space, ܻ a Banach lattice, ࣝ a collection of continuous convex operators ܲ: ܺ ՜ ܻ, ܲ() = , such that for any א ݔ ܺ, we have ݑݏԡܲ()ݔԡ < λ. Then the following relation ࣝא

holds:

ݑݏ

ԡܲ()ݔԡ < λ.

ࣝא,ԡ௫ԡ ஸଵ

In the sequel, ܺ will be an (F) space, i.e. a metrizable complete (not necessarily locally convex) topological vector space, ܻ will be a normed

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117

vector lattice (in particular, its norm is monotone on ܻା : ( ݕଵ ݕଶ ֜ ԡݕଵ ԡ ԡݕଶ ԡ ) and the multiplication with scalars is continuous). Recall that a normed vector lattice ܻ is a vector lattice endowed with a solid norm (|ݕଵ | |ݕଶ | ֜ ԡݕଵ ԡ ԡݕଶ ԡ ), such that the lattice operations are continuous. Consider a class ࣭ of sublinear operators Ȱ: ܺ ՜ ܻା such that Ȱ( = )ݔȰ(െܺ א ݔ )ݔ, Ȱ ࣭ א. Corollary 2.3. Let X, ܻ, ࣭ be as above. Assume that Ȱ is continuous Ȱ א ࣭ and ݑݏԡȰ()ݔԡ < λ ܺ א ݔ. Then there exists a convex closed ࣭א

neighborhood ܷ of 0 such that ڂ ࣭אȰ (ܷ) ܤ ؿଵ, ؔ ܤଵ (0 ), where ܤଵ (0 ) is the closed unit ball centered at the origin of the space ܻ. The poof follows the ideas from that of Theorem 2.1, also applying Baire’s theorem. Remark 2.1. Under previous conditions, assuming that ܻ is a normed vector lattice (the norm on ܻ is solid and the lattice operations are continuous), Corollary 2.3 says, in particular, that ݔଵ െ ݔଶ | ֜ ܷ אȰ(ݔଵ ) െ Ȱ(ݔଶ )| Ȱ(ݔଵ െ ݔଶ ) ܤ אଵ, ֜ Ȱ(ݔଵ ) െ Ȱ(ݔଶ ) ܤ אଵ, Ȱ ࣭ א It results that ࣭ is equicontinuous. Example 2.1. Using the above notations, let ࣦ be a family of linear continuous operators from ܺ to ܻ such that ݑݏԡܶ()ݔԡ < λ ܺ א ݔ. ்ࣦא

Define Ȱ( = )ݔȰ ் (|)ݔ(ܶ| ؔ )ݔ, ܺ א ݔ, ܶ ࣦ א. Then the family ࣭ = {Ȱ ் } ் ࣦאverifies the condition ݑݏԡȰ ் ()ݔԡ < λ ܺ א ݔ. ்ࣦא

Remark 2.2. In particular, Theorem 2.1 holds true when ܺ is a Banach space, ܻ is a normed vector lattice and the other conditions of Theorem 2.1 are accomplished. It is possible that a similar result be true for more general spaces ܺ (involving the notion of a barreled TVS). However, only for a few spaces it can be easily proved that they are barreled spaces, without using Baire theorem. On the other side, for applications the most important spaces are Banach spaces, especially Banach lattices. Theorem 2.4. Let ܺ be a Banach space, ܻ an order complete normed vector lattice with strong order unit ݑ , such that ܤଵ, = [െݑ , ݑ ]. Let ࣭ be a

118

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class of sublinear operators with the properties mentioned in Corollary 2.2. Additionally assume that Ȱ( = )ݔȰ(െܺ א ݔ )ݔ, Ȱ ࣭ א. Then the relation ෩ ( ݑݏ = )ݔȰ()ݔ Ȱ

ܧ א ݔ

࣭א

෩ , such that Ȱ ෩ ( = )ݔȰ ෩ (െא )ݔ defines a sublinear Lipschitz operator Ȱ ܻା ܺ א ݔ. Proof. Application of Corollary 2.3 leads to the existence of a closed ball of sufficiently small radius > ݎ0 such that ԡݔԡ ֜ ݎȰ(ܤ א )ݔଵ, = [െݑ , ݑ ] Ȱ ࣭ א It results Ȱ ൬ݎ

ԡݔԡ ݔ ൰ ݑ ֞ Ȱ( )ݔ ݑ {\ܺ א ݔ0 }, Ȱ ( ࣭ א5.1) ԡݔԡ ݎ

Thus, according to (5.1), for any fixed ܺ א ݔthe set {Ȱ( ;)ݔȰ } ࣭ אis bounded from above in ܻ. Thanks to the hypothesis on order completeness of ܻ, there exists ෩ ( ݑݏ ؔ )ݔȰ( )ݔ Ȱ ࣭א

ԡݔԡ ݑ ݎ

ܺ א ݔ

(5.2)

෩ is sublinear and has the property Ȱ ෩ ( = )ݔȰ ෩ (െא )ݔ It is easy to see that Ȱ ෩ ܻା ܺ א ݔ. Next we prove the Lipschitz property of Ȱ. To do this, one uses ෩ , the fact that the norm of ܻ is monotone the subadditivity property of Ȱ on ܻା , as well as relation (5.2). Namely, the following implications hold ෩ (ݔଶ )ห Ȱ ෩ (ݔଵ െ ݔଶ ) ֜ ෩ (ݔଵ ) െ Ȱ ݔଵ , ݔଶ ܺ א, หȰ ෩ (ݔଵ ) െ Ȱ ෩ (ݔଶ )ฮ ฮȰ ෩ (ݔଵ െ ݔଶ )ฮ ብ ฮȰ =

ԡݔଵ െ ݔଶ ԡ ݎ

ԡݔଵ െ ݔଶ ԡ ݑ ብ ݎ

෩ is a Lipschitz mapping from ܺ to ܻା . This concludes the proof. Ƒ Hence Ȱ

From Linear to Sublinear and to Convex Operators

119

Remark 2.3. Under the hypothesis of Theorem 2.4, each element of Ȱ ࣭ א is a Lipschitz operator, with the same Lipschitz constant 1Τݎ. Remark 2.4. It seems that topological completeness of ܻ is not necessary for the above results. However, the usual concrete spaces verifying the hypothesis of Theorem 2.4 are Banach spaces. Remark 2.5. The set ࣝ of all continuous sublinear operators Ȱ from ܺ to ܻା , such that Ȱ( = )ݔȰ(െܺ א ݔ )ݔ, Ȱ ࣝ א, ݑݏԡ߮()ݔԡ < λ א ݔ ఝࣝא

ܺ, is a convex cone. With the notations and under the assumptions of Theorem 2.4, the subset of all Ȱ ࣝ אformed by all elements of ࣝ with the property ߮൫ܤଵ, ൯ ܤ ؿଵ, , is convex, and its elements are the nonexpansive operators from ࣝ. If ݎfrom the proof of Theorem 2.4 is strictly greater than ෩ ), are contractions. 1, then the elements of ࣭, (as well as the operator Ȱ Remark 2.6. An arbitrary sublinear operator ߮: ܺ ՜ ܻା is a Lipschitz operator if and only if Ȱ is continuous at 0 . Corollary 2.5. Let X, Y be as in Theorem 2.4, ࣭ = {Ȱ ; ݊ }ߋ אa countable set of sublinear continuous operators from ܺ to ܻ, such that Ȱ (= )ݔ Ȱ (െܧ א ݔ )ݔ, א ݊Գ, and ݑݏԡȰ ()ݔԡ < λ ܺ א ݔ. Then the אԳ

relation ෩ ( ݑݏ = )ݔȰ ()ݔ Ȱ

ܺ א ݔ

אԳ

defines a sublinear Lipschitz operator ෩ (െܺ א ݔ )ݔ. Ȱ

෩ : ܺ ՜ ܻା , such that Ȱ ෩ (= )ݔ Ȱ

Corollary 2.6. Let ܺ, ܻ be as in Theorem 2.4, ࣮ = {Ȱ ; ݊ ߋ א, ݊ 1} a countable set of sublinear continuous operators from ܺ to ܻା , such that Ȱ ( = )ݔȰ (െܺ א ݔ )ݔ, { א ݊1,2, … }, and

ݑݏะ Ȱ ()ݔะ < λ ܺ א ݔ. אԳ, ஹଵ

Then the relation

ୀଵ

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Chapter Five

෩ ( ݑݏ = )ݔቌ Ȱ ()ݔቍ Ȱ אԳ, ஹଵ

ܺ א ݔ

ୀଵ

෩ : ܺ ՜ ܻା , such that Ȱ ෩ (= )ݔ defines a sublinear Lipschitz operator Ȱ ෩ (െܺ א ݔ )ݔ. Ȱ Example 2.2. Let ܭbe a compact Hausdorff topological space, ߤ a regular Borel probability measure on ܭ, ܺ ؔ )ܭ(ܥthe space of all real valued, continuous functions on ܭ, ܻ ؔ ݈ஶ the space of all bounded sequences of real numbers. The norm ԡήԡ௦௨ on the space ܺ is the sup-norm and the norm on ԡήԡ is the usual norm ԡήԡ = ԡήԡஶ , ԡ(ݔ )ஹଵ ԡஶ = sup|ݔ |. The space ஹଵ

ܻ = ݈ஶ verifies the hypothesis of Theorem 2.4, since it is an order complete normed vector lattice, the appropriate strong order unit being the sequence ݑ which has all the terms equal to 1. Define the scalar valued norms on ܺ ଵΤ

ܰ (݂) ؔ ቌන |݂| ݀ߤቍ

, ݂ ܺ א, ݇ אԳ, ݇ 1,

and the finite dimensional vector-valued norms on ܺ ܵ (݂) ؔ ԡ݂ԡ : ܺ ՜ ܻ, ԡ݂ԡ ؔ ൫ܰଵ (݂), 2ଵΤଶ ܰଶ (݂), … , ݊ଵΤ ܰ (݂), 0, … ,0, … ൯, ݊ אԳ, ݊ 1, ݂ ܺ א, ଵΤ

ܰ (݂) ԡ݂ԡ௦௨ ൫ߤ()ܭ൯

= ԡ݂ԡ௦௨ , ܰ (¶) = 1 ֜

sup

ԡԡೞೠ ୀଵ

ܰ (݂) = 1,

݇ { א1,2, … }, ݂ ܺ א. Consider the elementary function ݐ՜ ݃()ݐ(݈݊ ؔ )ݐ/ݐ, [ א ݐ1, λ), which is increasing on [1, ݁] and decreasing on the interval [݁, λ). This function has a global maximum point at ݐ = ݁ ( א2,3). It results that the function ݄: (1, λ) ՜ (0, λ), ݄( ݐ ؔ )ݐଵΤ௧ = ݁ (௧)Τ௧ has the same monotonicity properties, hence

From Linear to Sublinear and to Convex Operators

121

max ݇ ଵΤ ݉ܽݔ൛2ଵΤଶ , 3ଵΤଷ ൟ = 3ଵΤଷ { א ݊1,2, … }

ଵஸஸ

Thus, we obtain ݂ ֜ ܺ אS (݂) = ԡ݂ԡ max ݇ ଵΤ ԡ݂ԡ௦௨ ݑ 3ଵΤଷ ԡ݂ԡ௦௨ ݑ ݊ ଵஸஸ

{ א1,2, … } ֜ ෩ (݂) = ݑݏS୬ (݂) = ቀ݊ଵΤ ܰ (݂)ቁ Ȱ אԳ

3ଵΤଷ ή ԡ݂ԡ௦௨ ݑ , ݑ

ஹଵ

= (1, … ,1, … ), ݂ ܺ א, ෩ is the sublinear operator from Corollary 2.5. Observe that Ȱ ෩ has where Ȱ ଵΤଷ > 1. Next we apply the same method, replacing as Lipschitz constant 3 ݊ଵΤ by ݊ିଵΤ = ݁(ݔെ ݈݊(݊)Τ݊) 1 { א ݊1,2, … }. In this case, the above evaluations change into the following ones ෩ (݂) = ݑݏS (݂) = ቀ݊ିଵΤ ܰ (݂)ቁ Ȱ אԳ

ஹଵ

ԡ݂ԡ௦௨ ݑ ֜ ܺ א ݂

෩ (݂) െ Ȱ ෩ (݃)ห Ȱ ෩ (݂ െ ݃) ԡ݂ െ ݃ԡ௦௨ ݑ ֜ หȰ ෩ (݂ െ ݃)ฮ ԡ݂ െ ݃ԡ௦௨ ݂, ݃ ܺ א. ෩ (݂) െ Ȱ ෩ (݃)ฮ ฮȰ ฮȰ ி ෩ is a nonexpansive vector valued norm from ܺ To conclude, in this case Ȱ ෩ , consider to ܻ. To obtain contractions Ȱ (ܿ )ஹଵ ݈ = ܻ אஶ , 0 ܿ < ݍ1 ݊ 1, ܵ (݂) = (ܿଵ ܰଵ (݂), … , ܿ ܰ (݂), 0, … ,0, … ), ෩ (݂) = sup ܵ (݂) = ൫ܿ ܰ (݂)൯ Ȱ ݍԡ݂ԡ௦௨ ݑ ֜ ܺ א ݂ ஹଵ ஹଵ

෩ (݂) െ Ȱ ෩ (݃)ห Ȱ ෩ (݂ െ ݃) ݍԡ݂ െ ݃ԡ௦௨ ݑ ֜ หȰ ෩ (݂) െ Ȱ ෩ (݃)ฮ ݍԡ݂ െ ݃ԡ௦௨ ԡݑ ԡ = ݍԡ݂ െ ݃ԡ௦௨ ݂. ݃ ܺ א. ฮȰ

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Chapter Five

෩ : ܺ ՜ ܻା is a contraction vector-valued norm, of contraction Thus Ȱ constant ݍ, and the best value for ݍis = ݍsup ܿ . In particular, if 0 ஹଵ

෩ is a contraction operator, of inf ܿ sup ܿ = 1Τ2 ݊ 1, then Ȱ

ஹଵ

ஹଵ

contraction constant = ݍ1Τ2. In this example, the operators Ȱ mentioned in Corollary 2.6 stand for (0, … ,0, ܿ ܰ (݂), 0,0, … ), and ܿ ܰ (݂) is the ݊ െ ݄ݐcoordinate of the vector ܵ (݂) ܻ אା .

3. Relationship between linear and sublinear continuous operators In Theorem 3 proved below, we point out some remarks related on fixed points (if any) of some special sublinear operators. Theorem 3.1. Let ܺ be an order complete normed vector lattice, Ȱ: ܺ ՜ ܺା a sublinear operator which is continuous at and verifies the conditions Ȱ( = )ݔȰ(െܺ א ݔ )ݔ. Assume that there exists a fixed point ݔ ് of Ȱ. Denote by ࣦ the set of all linear operators from ܺ to ܺ, with the following properties ܶ(ݔ ) = ݔ , ܶ( )ݔ Ȱ(ܺ א ݔ )ݔ. Define Ȱ ் (|)ݔ(ܶ| ؔ )ݔ, ܺ א ݔ, ܶ ࣦ א . Then ࣭ ؔ {Ȱ ் ; ܶ ࣦ א } is a nonempty equicontinuous family of sublinear operators, for which the following relations hold true Ȱ ் (ݔ ) = ݔ , Ȱ ் ( )ݔ Ȱ(ܺ א ݔ )ݔ, ࣦ א ܶ

(5.3)

Proof. According to the Hahn-Banach theorem, ࣦ is nonempty, hence so is ࣭. We next prove that ࣦ is equicontinuous. Continuity of Ȱ at 0 leads to the existence of a sufficiently small > ݎ0 such that ԡݔԡா ֜ ݎ ௫ ԡȰ()ݔԡ 1. This further yields ቛȰ ቀ ݎቁቛ 1 ܺ א ݔ, ് ݔ0 , ԡ௫ԡ ଵ

which can be rewritten as ԡȰ()ݔԡ ԡݔԡ ؔ ܿԡݔԡ, ܺ א ݔ. On the other side, ܶ( )ݔ Ȱ( )ݔ(ܶ ֜ ܺ א ݔ )ݔ Ȱ()ݔ, െܶ((ܶ = )ݔെ )ݔ Ȱ(െ)ݔ = Ȱ(֜ ܺ א ݔ )ݔ Ȱ ் ( |)ݔ(ܶ| = )ݔ Ȱ(ܺ א ݔ )ݔ

(5.4)

(recall that the range of Ȱ is contained in ܧା ). Since the norm on ܺ is solid, ଵ we infer that ԡܶ()ݔԡ ԡȰ()ݔԡ ܿԡݔԡ ܺ א ݔ, ܿ ؔ , ࣦ א ܶ . It results ԡܶԡ ܿ ࣦ א ܶ , so that ࣦ is equicontinuous. It is also a convex

From Linear to Sublinear and to Convex Operators

123

subset of the space )ܧ(ܤof all bounded linear operators applying ܧto itself. Here ԡήԡ denotes the usual operatorial norm on )ܧ(ܤ. To finish the proof, we observe that Ȱ(ݔ ) ܺ אା implies Ȱ ் (ݔ ) = |ܶ(ݔ )| = |ݔ | = |Ȱ(ݔ )| = Ȱ(ݔ ) = ݔ ࣦ א ܶ . Thus, the first property (5.3) is verified. The second property (5.3) has already been proved by (5.4). Using (5.4) once more and the inequality ԡȰ()ݔԡா ܿԡݔԡா ܺ א ݔ, we derive ԡȰ ் ()ݔԡ ԡȰ()ݔԡ ܿԡݔԡ ܺ א ݔ

(5.5)

Now the subadditivity of each Ȱ ் and (5.5) lead to |Ȱ ் (ݔଵ ) െ Ȱ ் (ݔଶ )| Ȱ ் (ݔଵ െ ݔଶ ) ֜ ԡȰ ் (ݔଵ ) െ Ȱ ் (ݔଶ )ԡ ԡȰ ் (ݔଵ െ ݔଶ )ԡ ܿԡݔଵ െ ݔଶ ԡ , ݔଵ , ݔଶ ܺ א, ࣦ א ܶ. This proves the equicontinuity of ࣭ DQGFRQFOXGHVWKHSURRIƑ The next two results follow via Krein-Milman Theorem. Proposition 3.2. Let ܺ be a reflexive Banach space endowed with a linear order relation defined by a closed positive cone ܺା , and ܺ ؿ ܤା a convex bounded closed subset such that any ܺ א ݔା \{ } can be represented uniquely as ܾߩ = ݔfor some ߩ ( א0, λ) and ܾ ܤ( ܤ אis a base for ܺା ). Assume that ܻ is a topological vector space endowed with an order relation defined by a closed positive cone ܻା , Ȱ: ܺ ՜ ܻ is a bounded sublinear operator, and ܶ ܺ(ܤ א, ܻ) is a bounded linear operator such that Ȱ(݁) ܶ(݁) )ܤ(ݎݐݔܧ א ݁ Then Ȱ( )ݔ ܶ(ܺ א ݔ )ݔା . Proof. Clearly, ܤis weakly compact and convex, so that = ܤ ݈ܿ ቀܿ൫)ܤ(ݎݐݔܧ൯ቁ. The topological closure of the convex set ܿ൫)ܤ(ݎݐݔܧ൯ in the weak topology equals its topological closure in the norm topology on ܺ. Let

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ݔ = σୀଵ ߙ ݁ ܿ א൫)ܤ(ݎݐݔܧ൯, ߙ [ א0, λ), σୀଵ ߙ = 1, ݁ א )ܤ(ݎݐݔܧ, ݆ = 1, … , ݊. Then by the assumptions of the statement we derive

Ȱ(ݔ ) ߙ Ȱ൫݁ ൯ ߙ ܶ൫݁ ൯ = ܶ ቌ ߙ ݁ ቍ = ܶ(ݔ ), ୀଵ

ୀଵ

ୀଵ

݊ אԳ, ݊ 1. For ܾ ݈ܿ אቀܿ൫)ܤ(ݎݐݔܧ൯ቁ = ܤ, ܾ = lim ݔ , ݔ ܿ א൫)ܤ(ݎݐݔܧ൯ for all

݊ 1, the continuity of Ȱ,ܶ, as well as the hypothesis that the positive cone of ܻ is closed, lead to Ȱ(ܾ) = lim Ȱ(ݔ ) lim ܶ(ݔ ) = ܶ(ܾ).

Now let ܺ א ݔା \{ }, ܾߩ = ݔ, ߩ > 0, ܾ ܤ א. Then Ȱ( = )ݔȰ(ߩܾ) = ߩȰ(ܾ) ߩܶ(ܾ) = ܶ(ߩܾ) = ܶ()ݔ. 7KLVHQGVWKHSURRIƑ Proposition 3.3. Under the hypothesis of Proposition 3.2, additionally assume that ܧ, ܨare normed vector lattices (their norms are solid), and Ȱ is isotone. Then ԡȰԡ ԡܶԡ. Proof. According to Proposition 3.2, we have already seen that Ȱ( )ݓ ܶ( )ݓfor all ܺ א ݓା . Using this and monotonicity of Ȱ, we derive (Ȱ( )ݔ Ȱ(| )|ݔ ܶ(|)|ݔ, െȰ( )ݔ Ȱ(െ )ݔ Ȱ(| )|ݔ ܶ(|֜ ))|ݔ |Ȱ( |)ݔ ܶ(| ֜ )|ݔԡȰ()ݔԡ ԡܶԡԡݔԡ ֜ ܺ א ݔԡȰԡ ԡܶԡ. 7KLVFRQFOXGHVWKHSURRIƑ

CHAPTER SIX MOMENT PROBLEM, FINITE-SIMPLICIAL SETS AND RELATED SANDWICH RESULTS

1. Introduction The results of this chapter are based on the papers [2] and [59]. They were also published in [58]. We start by recalling that Theorem 2.32 of Chapter 1was applied in [2] to deduce a sandwich type result of intercalating an affine functional ݄: ܵ ՜ Թ between a convex functional ݂: ܺ ՜ Թ and a concave functional ݃: ܵ ՜ Թ, ݂ ݃, where ܵ is a finite simplicial subset of a real vector space ܺ. Here the novelty is that a finite simplicial set ܵ may be unbounded in any locally convex topology on ܵ (see Section 2 below and the paper [2]). In the present chapter, we prove a topological version of the sandwich result mentioned above (see Theorem 2.3 below). Some other related results are recalled or respectively proved. It is worth noticing that, unlike the direct Hahn-Banach theorems, here the dominating functional ݃ is concave, while the minorating functional ݂ is convex. A previous such sandwich result for real functions defined on a Choquet simplex ܺ is recalled in [65] and in Section 2 below. The rest of this chapter is organized as follows. In Section 2, topological versions for known sandwich results as those mentioned above are proved. Related remarks are discussed. Section 3 is devoted to extending inequalities from smaller subsets to the entire positive cone ܺା , by means of Krein-Milman and respectively Carathéodory’s theorem. Section 4 (the last one) refers to remarks on the subdifferentials of continuous sublinear operators.

2. Finite simplicial sets and related sandwich theorems We start by a preliminary lemma. Recall that a base ܤof a convex cone ܥ contained in a vector space ܺ is a set of the form ܪ ת ܥ = ܤ, where ܪis a hyperplane which misses the origin, defined by a strictly positive linear functional

126

Chapter Six

߮: ܺ ՜ Թ (߮( > )ݔ0 for all ܥ א ݔ, ് ݔ). If such a functional does exist, then choose any real ߚ > 0 and define }ߚ = )ݔ(߮ ;ܺ א ݔ{ = ܪ. Then for any ܥ א ݔ, ് ݔ0, there exists a unique ܾ ܥ ת ܪ = ܤ אand a unique real number ߙ > 0 such that ܾߙ = ݔ Indeed, if ܾ = ߚ

௫

, then ܾ ܤ אand

ఝ(௫)

=ݔ

߮()ݔ ܾ = ߙܾ. ߚ

For the first result (Lemma 2.1), we will restrict ourselves to the Banach lattice setting, since the topological properties involved are important. In this respect, the hyperplane ܪand the positive cone ܧା are closed subsets (߮ is continuous). Lemma 2.1. Let ܺ, ܻ be Banach lattices. Assume that the positive cone ܺା has a base ܺ ת ܪ = ܤା , where ܪis a closed hyperplane missing the origin, as mentioned above. Let ܲ, െܳ: ܤ՜ ܻ be convex continuous bounded operators on ܤ and ݄: ܤ՜ ܻ an affine continuous bounded operator. Define Ȱ, Ȳ: ܧା ՜ ܨ, Ȱ(ߣܾ) = ߣܲ(ܾ), Ȳ(ߣܾ) = ߣܳ(ܾ), ܾ ܤ א, ߣ אԹା . Then Ȱ is a sublinear continuous extension of ܲ, Ȳ is supralinear, continuous and extends ܳ, while ܶ(ߣܾ) = ߣ߮(ܾ) defines an additive positively homogeneous operator on ܺା , which can be further extended to a linear continuous operator ܶ from ܺ to ܻ. In addition, we have

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ܲ ݄ ܳ on ܤif and only if Ȱ ܶ Ȳ on ܺା . ܗܗܚ۾. By definition, Ȱ, Ȳ, ܶ extend ܲ, ܳ, ߮ respectively, from ܤto the entire positive cone ܧା . To prove that Ȱ is subadditive, let ݔଵ = ߣଵ ܾଵ , ݔଶ = ߣଶ ܾଶ be two elements in the positive cone ܺା , where: ܾ ܤ א, ߣ [ א0, λ), ݆ = 1,2, ߣଵ + ߣଶ > 0. Then ߣଵ ߣଶ ܾ + ܾ ൰൱ ؔ ߣଵ + ߣଶ ଵ ߣଵ + ߣଶ ଶ ߣଵ ߣଶ (ߣଵ + ߣଶ )ܲ ൬ ܾ + ܾ ൰ ߣଵ + ߣଶ ଵ ߣଵ + ߣଶ ଶ ߣଵ ߣଶ (ߣଵ + ߣଶ ) ൭ ܲ(ܾଵ ) + ܲ(ܾଶ )൱ = ߣଵ + ߣଶ ߣଵ + ߣଶ

ߔ(ݔଵ + ݔଶ ) = ߔ ൭(ߣଵ + ߣଶ ) ൬

ߣଵ ܲ(ܾଵ ) + ߣଶ ܲ(ܾଶ ) = Ȱ(ݔଵ ) + Ȱ(ݔଶ ) Next, we prove that Ȱ is positively homogeneous: if ܾߣ = ݔ, ܾ ܤ א, ߣ א (0, λ), ߙ ( א0, λ), then Ȱ(ߙ = )ݔȰ൫(ߙߣ)ܾ൯ = (ߙߣ)ܲ(ܾ) = ߙ൫ߣܲ(ܾ)൯ = ߙȰ()ݔ Thus Ȱ is sublinear, Ȳ is supralinear and ܶ is additive and positively homogeneous on the positive cone of ܺ since ݄ is assumed to be simultaneous convex and concave on ܤ. Extend ܶ to the space ܺ = ܺା െ ܺା by ܶ(ݔଵ െ ݔଶ ) ؔ ܶ(ݔଵ ) െ ܶ(ݔଶ ), where ݔ ܺ אା , ݆ = 1,2 The definition makes sense, since it does not depend on decomposing ݔas a difference of two arbitrary elements from the positive cone ܧା . Indeed, if ݔଵ െ ݔଶ = ݕଵ െ ݕଶ , then ݔଵ + ݕଶ = ݔଶ + ݕଵ , ݔ , ݕ ܺ אା , ݆ = 1,2. Now additivity of ܶ on ܺା yields

128

Chapter Six

ܶ(ݔଵ ) + ܶ(ݕଶ ) = ܶ(ݔଶ ) + ܶ(ݕଵ ) ֞ ܶ(ݔଵ ) െ ܶ(ݔଶ ) = ܶ(ݕଵ ) െ ܶ(ݕଶ ) For each nonnegative scalar ߣ, clearly it results ܶ(ߣݔߣ(ܶ = )ݔଵ ) െ ܶ(ߣݔଶ ) = ߣ൫ܶ(ݔଵ ) െ ܶ(ݔଶ )൯ = ߣܶ()ݔ, ݔ ܺ אା , ݆ = 1,2. On the other hand, according to the definition from above, we infer that ܶ(െ(ܶ = )ݔെ(ݔଵ െ ݔଶ )) = ܶ(ݔଶ ) െ ܶ(ݔଵ ) = െ൫ܶ(ݔଵ ) െ ܶ(ݔଶ )൯ = െܶ()ݔ. If ߣ < 0, then ߣ = െߩ, where ߩ = െߣ > 0. From the preceding remark, we derive ܶ(ߣ(ܶ = )ݔെߩ(ܶߩ = )ݔെ = )ݔെߩܶ()ݔ(ܶߣ = )ݔ. To finish the proof, we only have to prove the assertions on the continuity. If (ݔ )ஹଵ is a sequence of elements from ܺା , such that ݔ = ߣ ܾ ՜ ܾߣ = ݔ, ܾ , ܾ ܤ א, ߣ , ߣ > 0, then the following relations hold true ܶ(ݔ ) = ߣ ߮(ܾ ) = ߣ ߚ ՜ ߮(ߚߣ = )ݔ, since ߮ is continuous on ܺ. Thus ߣ ՜ ߣ. These relations lead to: ܾ =

1 1 ݔ ՜ ܾ = ݔ. ߣ ߣ

Due to the continuity of ܲ on ܤ, it results ܲ(ܾ ) ՜ ܲ(ܾ), which further implies Ȱ(ݔ ) = ߣ ܲ(ܾ ) ՜ ߣܲ(ܾ) = Ȱ()ݔ. A special case is = ݔ0, when ݔ = ߣ ܾ ՜ ֜ ߣ ߚ = ߮(ݔ ) ՜ 0 ֜ ߣ ՜ 0 ֜

Moment Problem, Finite-Simplicial Sets and Related Sandwich Results

129

Ȱ(ݔ ) = ߣ ܲ(ܾ ) ՜ 0, since ܲ is bounded on ܤby hypothesis. Thus the continuity of Ȱ, Ȳ and ܶ on ܧା is proved. We only have to show that the linear extension of ܶ to the whole space ܺ is continuous, where this extension will be denoted by ܶ too. Clearly, the continuity at the origin will be sufficient. The following implications hold true: ݔ ՜ 0 ֜ ݔା ՜ 0 ֜ ܶ(ݔା ) ՜ 0. Similarly, ܶ(ݔି ) ՜ 0, hence T(ݔ ) = ܶ(ݔା ) െ ܶ(ݔି ) ՜ 0. Finally, observe that ܲ ݄ ܳ on = )ܾ(ܲߣ ֞ ܤȰ(ߣܾ) ߣ݄(ܾ) = ܶ(ߣܾ) ߣܳ(ܾ) = Ȳ(ߣܾ), ߣ > 0, ܾ ֞ ܤ א Ȱ( )ݔ ܶ( )ݔ Ȳ()ݔ, ܺ א ݔା . 7KHSURRILVFRPSOHWHƑ Next, we recall the definition of a simplex in an infinite dimensional locally convex space ܺ, and we emphasize one of its main properties. If ܥis a convex cone in ܺ, ܥhas a base ܤand the linear order relation defined by ܥ on ܺଵ = ܥെ ܥis laticial, then ܤis called a Choquet simplex. Usually, a simplex is assumed to be also compact. For more information on simplexes see [65]. Now we recall the statement of D.A. Edwards’ separation theorem (see [65, Theorem 16.7]). Lemma 2.2 (Edwards). If ݂ and െ݃ are convex upper semicontinuous real valued functions on a simplex ܤcontained in a locally convex space, with ݂ ݃, then there exists a continuous affine function ݄ on ܤsuch that ݂ ݄ ݃. The next result is a version of Lemma 2.2, formulated on a cone having a simplex base. Theorem 2.3. Let ܺ be a locally convex space, ܥa convex cone in ܺ which has a simplex ܤas a base. Assume that the trace of the topology of ܺ on ܺଵ = ܥെ ܥis locally solid with respect to the order relation defined by ܥ. Let ߔ: ܥ՜ Թ be a continuous sublinear functional, ߖ: ܥ՜ Թ a continuous

130

Chapter Six

supralinear functional such that ߔ(݁) = ߖ(݁) for all ݁ )ܤ(ݎݐݔܧ א. Then there exists a unique continuous linear functional ܶ: ܧଵ ՜ Թ such that ߔ( )ݔ ܶ( )ݔ ߖ(ܥ א ݔ )ݔ

(6.1)

Proof. Let ݂ ؔ ߔ| , ݃ ؔ ߖ| . Then ݂ is convex and continuous, ݃ is concave and continuous, and ݂ ݃ on ݂( ܤെ ݃ is continuous and convex, vanishing on )ܤ(ݔܧby hypothesis; it results (݂ െ ݃)( )ݔ 0 for all א ݔ ܿ൫)ܤ(ݎݐݔܧ൯ and via continuity, (݂ െ ݃)( )ݔ 0 for all א ݔ ݈ܿ ቀܿ൫)ܤ(ݎݐݔܧ൯ቁ = ܤ, where the last equality is given by Krein-Milman theorem). Since ܤis a simplex, application of Lemma 2.2 leads to the existence of a continuous affine function ݄: ܤ՜ Թ, such that ݂ ݄ ݃. According to the first part of the proof of Lemma 2.1, ݄ has a unique linear extension, say ܶ, to ܧଵ = ܥെ ܥ, such that (6.1) holds true. The next step is to prove the continuity of ܶ on ܧଵ , which is equivalent to its continuity at the origin. Let (ݔఋ )ఋאο be a generalized sequence in ܧଵ , such that ݔఋ ՜ 0. Consider the sequences (ݔఋା )ఋאο , (ݔఋି )ఋאο in ܥ. According to the assumptions on the topology on ܧଵ , also using the continuity at the origin of ߔ, ߖ, and (6.1) as well, we infer that ݔఋ ՜ 0 ֜ (ݔఋା ՜ 0, ݔఋି ՜ 0) ֜ ( ߔ(ݔఋା ) ՜ 0, ߖ(ݔఋା ) ՜ 0, ߔ(ݔఋା ) ܶ(ݔఋା ) ߖ(ݔఋା )) ֜ ܶ(ݔఋା ) ՜ 0 Similarly, ܶ(ݔఋି ) ՜ 0, so that ܶ(ݔఋ ) = ܶ(ݔఋା ) െ ܶ(ݔఋି ) ՜ 0. Hence ܶ is continuous. To finish the proof, we only have to show the uniqueness of ܶ with the properties in the statement. Let ܶଵ be a linear continuous functional on ܧଵ such that (6.1) holds for ܶଵ instead of ܶ. Let ݄ଵ be the restriction of ܶଵ to the simplex ܤ. Then ݄ଵ is continuous and affine on ܤ. Moreover, we have ݄ଵ (݁) = ܶଵ (݁) )݁(ߔ[ א, ߖ(݁)] = {ܶ(݁)} = {݄(݁)} ݁ ֜ )ܤ(ݎݐݔܧ א ݄ଵ (݁) = ݄(݁) )ܤ(ݎݐݔܧ א ݁

Moment Problem, Finite-Simplicial Sets and Related Sandwich Results

131

This further yield ݄ଵ ( )ݔ(݄ = )ݔfor all ܿ א ݔ൫)ܤ(ݎݐݔܧ൯ (via the property of being affine for ݄, ݄ଵ ). Now also using the continuity of the two involved functions, we derive ݄ଵ ( )ݔ(݄ = )ݔfor all ݈ܿ א ݔቀܿ൫)ܤ(ݎݐݔܧ൯ቁ = ܤ, where the last equality is a consequence of Krein-Milman theorem. Thus ܶଵ | = ܶ| , which means ܶଵ = ܶ on ܺଵ . 7KLVHQGVWKHSURRIƑ Remark 2.1. Let ܺ = ݈ଵ and ܻ an arbitrary Banach space endowed with a linear order relation such that the positive cone ܻା is closed. Let {݁ }ஶ ୀଵ be the canonical base of ܺ and {ݕ }ஶ ୀଵ a bounded sequence in ܻ. Assume that Ȱ, െȲ: ܺା ՜ ܻ are given sublinear continuous operators such that Ȱ(݁ ) = Ȳ(݁ ) for all natural numbers ݊ 1. Then there exists a unique bounded linear operator ܶ ܺ(ܤ א, ܻ) such that Ȱ T|శ Ȳ. Indeed, if we denote ݕ = Ȱ(݁ ), ݊ 1 and define ஶ

ܶ( = )ݔ ݔ ݕ , ݔ( = ݔ )ஹଵ ܺ א,

(6.2)

ୀଵ

then the series on the right-hand side is absolutely convergent in the Banach space ܻ for each fixed element ܺ א ݔ. Thus (6.2) defines a linear operator, which is continuous, with ԡܶԡ = ݑݏԡݕ ԡ < λ ஹଵ

From (6.2), also using the hypothesis, it results ܶ(݁ ) = ݕ = Ȱ(݁ ) = Ȳ(݁ ), ݊ 1. This and the continuity of all involved operators, as well as sublinearity of Ȱ, െȲ lead to

132

Chapter Six

Ȱ( )ݔ T( )ݔ Ȳ()ݔ for all ܺ א ݔା . It is possible that such examples could be adapted for more general spaces ܺ. In the end of this section, we recall that related sandwich theorems hold on the finite simplicial sets, as discussed in [2]. A convex subset ࣝ of the vector space ܺ is called finite simplicial if for any finite dimensional convex compact subset ࣝ ؿ ܭ, there exists a finite dimensional simplex ࣭ such that ࣝ ؿ ࣭ ؿ ܭ. Here the novelty is that ࣝ is not supposed to be bounded in a locally convex topology on ܺ. Here are a few examples. 1) In Թ , ݊ 2, any convex cone ࣝ having a base that is a simplex (the corresponding order relation is laticial) is an unbounded finite simplicial set. 2) In Թ , ݊ 2, for each ( א 1, λ), the convex cone ܺ defined by ଵΤ

ିଵ

ࣝ = ൞(ݔଵ , … , ݔ ); ݔ ቌหݔ ห ቍ

ൢ

ୀଵ

has a compact base, but ࣝ is not finite-simplicial. 3) Let ܺ be an arbitrary infinite or finite dimensional vector space (of dimension 2), ܶ: ܺ ՜ Թ a non-null linear functional and ߙ אԹ. Then the sets ࣝଵ = { )ݔ(ܶ ;ݔ ߙ}, ࣝଶ = { )ݔ(ܶ ;ݔ ߙ} are finitesimplicial. 4) Let ܺ, ܶ be as in Example 3), ߙ, ߚ two real numbers such that ߙ < ߚ. The set ࣝ = { ߙ ;ݔ ܶ( )ݔ ߚ} is not finite-simplicial. From the last two examples, we easily infer that generally the intersection of two finite-simplicial sets is not finitesimplicial. The following sandwich type result holds true (cf. [2]). Theorem 2.4. Let be an arbitrary vector space, ࣝ a finite-simplicial subset, ݂: ࣝ ՜ Թ a convex functional, ݃: ࣝ ՜ Թ a concave functional such

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133

that ݂ ݃ on ࣝ. Then there exists an affine functional ݄: ࣝ ՜ Թ such that ݂ ݄ ݃. Here is a topological version of Theorem 2.2. Theorem 2.5. Let ܺ be an ordered Banach space. Assume that ܺା is finite simplicial and there exists ݔ ܺ אା such that ܺା െ ݔ contains a balanced, absorbing, convex subset. Let ݂, െ݃: ܺା ՜ Թ be convex continuous functions such that ݂ ݃. Assume also that ݂() = ݃() = 0. Then there exists a continuous linear functional ܶ: ܧ՜ Թ such that ݂ ܶ ݃ on ܺା . Proof. According to Theorem 2.4, there exists an affine functional ݄: ܺା ՜ Թ such that ݂ ݄ ݃ on ܺା . Using also the hypothesis, we infer that 0 = ݂() ݄(0) ݃() = 0, hence ݄() = 0. On the other hand, since ܺା െ ݔ is convex, absorbing and balanced, it is a neighborhood of in the finest locally convex topology on ܺ (see [69]). Recall that in this topology, any convex balanced absorbing subset is a neighborhood of (this is the topology generated by the family of all semi-norms on ܺ). Hence there exists a convex open neighborhood ܸ of in this topology, such that ݔ = ܦ + ܸ ܺ ؿା In particular, the interior ݅݊ܺ(ݐା ) of ܺା with respect to the finest locally convex topology on ܺ is not empty. Denote this interior by ܥ. Since ݄ is convex and concave on the convex open subset ( ܥin the finest locally convex topology on ܺ) and ݔ ܥ א, the subdifferentials of ݄ and െ݄ at

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ݔ are nonempty. This means that there exist linear functionals ܷ and ܹ on ܺ such that ܷ( )ݔെ ܷ(ݔ ) ݄( )ݔെ ݄(ݔ ) ܹ( )ݔെ ܹ(ݔ ) for all ܥ א ݔ. Since clearly any linear form on ܺ is continuous with respect to the finest locally convex topology, from the last two inequalities we infer that ݄ is continuous at ݔ . Repeating this argument for any ݔ ܥ א, we infer that ݄ is continuous on ܥwith respect to the topology under attention. As a consequence, the subsets ݔ({ = ܣ, × ܥ א )ݎԹ; ݄(}ݎ < )ݔ, ݔ({ = ܤ, × ܥ א )ݎԹ; ݄(}ݎ > )ݔ are open convex disjoint subsets in × ܥԹ endowed with the product topology. By separation theorem, there exists a hyperplane × ܧ ؿ ܪԹ, separating (strictly) the subsets ܣand ܤ. So, there is a not null linear form ܮon ܺ × Թ and ߙ אԹ such that ݔ({ = ܪ, × ܺ א )ݎԹ; ݔ(ܮ, }ߙ = )ݎ, ݔ({ ؿ ܣ, × ܥ א )ݎԹ; ݔ(ܮ, }ߙ < )ݎ, ݔ({ ؿ ܤ, × ܥ א )ݎԹ; ݔ(ܮ, }ߙ > )ݎ. All the topological notions in the sequel refer to the finest locally convex topology, unless other specification is mentioned. Since ܮis linear, it is continuous on ܺ × Թ. For (ݔ, ݎଵ ) ܣ א, we have ݔ(ܮ, 0) + ݎଵ (ܮ, 1) < ߙ, while for any (ݔ, ݎଶ ) ܤ א, it results ݔ(ܮ, 0) + ݎଶ (ܮ, 1) > ߙ. From these last two inequalities written for a ܥ א ݔand ݎଵ ് ݎଶ , we infer that ݔ(ܮ, 0) + ݎଵ (ܮ, 1) < ߙ < ݔ(ܮ, 0) + ݎଶ (ܮ, 1), that implies (ݎଶ െ ݎଵ )(ܮ, 1) > 0. In particular, it results (ܮ, 1) ് 0. Going back to the definitions of the subsets ܣand ܤ, observe that for any

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ܥ א ݔand any ߝ > 0 the element (ݔ, ݄( )ݔ+ ߝ) ܣ א, while (ݔ, ݄( )ݔെ ߝ) א ܤ. This remark yields: ݔ(ܮ, 0) + (݄( )ݔ+ ߳)(ܮ, 1) < ߙ, ݔ(ܮ, 0) + (݄( )ݔെ ߳)(ܮ, 1) > ߙ. Passing to the limit as ߝ ՜ 0, one obtains ݔ(ܮ, 0) + ݄((ܮ)ݔ, 1) = ߙ for all ܥ א ݔ. This may be written as ݄(= )ݔ

ݔ(ܮ, 0) ߙ െ . (ܮ, 1) (ܮ, 1)

Hence ݄ is the restriction to ܺ(ݐ݊݅ = ܥା ) of the linear form ܶ defined on ܺ by ܶ( = )ݔെ then adding the constant

ఈ (,ଵ)

(௫,) (,ଵ)

,

. Recall now that we have already proved that

݄() = 0, which yields ߙ = 0, so that ݄ is the restriction to ܥof the linear form ܵ. On the other hand, by hypothesis, ݂ and ݃ are assumed to be continuous with respect to the norm topology. In particular, they are continuous with respect to the finest locally convex topology and ܶ has this property as well. We also infer that ݂| ܶ ݃| and ܺ(ݐ݊݅ = ܥା ) is dense in ܺା . From the continuity of the three involved functionals, the inequalities written above are extended to the entire positive cone ܺା . To finish the proof, we must show that ܵ is continuous with respect to the norm topology, that is equivalent to its continuity at . From now on, only the norm topology is involved. Let ݔ ՜ . As it is known, the ordered Banach space ܺ can be renormed by means of an equivalent norm, such that it becomes a regularly ordered Banach space. In such a space for any sequence ݔ ՜ , there exists sequences (ݑ ) , (ݒ ) , ݑ , ݒ ܺ אା , ݔ = ݑ െ ݒ for all ݊ אԳ, such that ݑ ՜ , ݒ ՜ . It results 0 = ݂() ՚ ݂(ݑ ) ܶ(ݑ ) ݃(ݑ ) ՜ ݃() = 0

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which obviously imply ܶ(ݑ ) ՜ 0. Similarly, ܶ(ݒ ) ՜ 0, so that ܶ(ݔ ) = ܶ(ݑ ) െ ܶ(ݒ ) ՜ 0 7KLVFRQFOXGHVWKHSURRIƑ

3. Extending inequalities via Krein-Milman and Carathéodory’s theorems The next results extend an inequality occurring on a small set to a much larger subset. Theorem 3.1. Let ܺ be a reflexive Banach lattice, ܻ an order complete Banach lattice in which every topological bounded subset is order-bounded and ݕ ՛ ݕimplies ݕ ՜ ݕ, Ȱ: ܺା ՜ ܻ a quasiconvex continuous positively homogeneous operator, ܶ ܤ אା (ܺ, ܻ) a positive linear operator such that Ȱ(݁) ܶ(݁) for all extreme points ݁ of the set ܺ ؔ ܭା ܤځଵ, . Then Ȱ( )ݔ ԡݔԡ ή ݑݏȰ(݁) ܻ אା ܺ א ݔା . אா௫()

Proof. Recall that an operator Ȱ from a convex subset ܥof a vector space ܺ to a vector lattice ܻ is called quasiconvex if ߔ൫(1 െ ݔ)ݐଵ + ݔݐଶ ൯ ݔ(߮{ݑݏଵ ), ߮(ݔଶ )}, [ א ݐ0,1], ݔଵ , ݔଶ ܥ א. Following the proof (by induction) of Jensen’s inequality for real convex functions, for any convex combination σୀଵ ߙ ݔ , ݔ ܥ א, ݆ = 1, … , ݊, a quasiconvex operator Ȱ verifies

Ȱ ቆ ୀଵ

ߙ ݔ ቇ {ݑݏȰ(ݔଵ ), … , Ȱ(ݔ )}

(6.3)

See [38] for details, examples and exercises related to this important notion. The set ܺ ؔ ܭା ܤځଵ, is convex, weakly compact and ݈ܿ = ܭቀܿ൫)ܭ(ݎݐݔܧ൯ቁ holds thanks to the Krein-Milman theorem. Let

(6.4)

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137

ݔ =

ୀଵ

ߙ ݁ ܿ א൫)ܭ(ݎݐݔܧ൯, ݁ )ܭ(ݔܧ א,

ߙ [ א0, λ), ݆ = 1, … , ݊, ߙ = 1. ୀଵ

According to (6.3) and also using the hypothesis, we infer that Ȱ(ݔ ) {ݑݏȰ(݁ଵ ), … , Ȱ(݁ )} ݁(ܶ{ݑݏଵ ), … , ܶ(݁ )}

(6.5)

On the other side, any positive linear operator from ܺ to ܻ is continuous, so that the image of the bounded set ܺ ؿ ܭା through the positive (bounded) linear operator ܶ is topologically bounded, hence is െ bounded in ܻା . Thus it results ܶ([ ؿ )ܭ , ݕ ] for some ݕ ܻ אା From this and also using (6.5), it results Ȱ(ݔ ) ݕ , ݔ ܿ א൫)ܭ(ݎݐݔܧ൯, ݊ אԳ, ݊ 1. If = ݔlim ݔ ݈ܿ אቀܿ൫)ܭ(ݔܧ൯ቁ = ܭ, where ݔ ܿ א൫)ܭ(ݎݐݔܧ൯ for all ՜ஶ

݊ 1, then, thanks to the continuity of Ȱ, we are leaded to Ȱ( = )ݔlim Ȱ(ݔ ) lim ൫ݑݏ൛Ȱ൫݁ ൯; ݆ = 1, … , ݊ൟ൯ = ՜ஶ

ݑݏȰ(݁ ) ஹଵ

՜ஶ

ݑݏȰ(݁) אா௫௧()

)݁(ܶ ݑݏ ݕ , ܭ א ݔ.

אா௫௧()

Since Ȱ is positively homogeneous, application of this evaluation to ݔΤԡݔԡ ܭ א, for all ܺ א ݔା , ് ݔ0 , yields Ȱ( )ݔ ԡݔԡ ή

ݑݏȰ(݁) ԡݔԡݕ , ܺ א ݔା .

אா௫௧()

7KLVFRQFOXGHVWKHSURRIƑ Theorem 3.2. Let ܺ be an order complete normed vector lattice, ܺ ؿ ܭa finite dimensional compact subset, (Ȱ )ஹ a sequence of continuous sublinear operators from ܺ to ܺ, such that for each ܺ א ݔ, there exists ෩ ( ݈݉݅ ؔ )ݔȰ (ܺ א )ݔ. Assume that for each ݊ אԳ, there exists an Ȱ ՜ஶ

138

Chapter Six

affine operator ܶ from ܺ to ܺ, such that ܶ ( )ݔ Ȱ ( ܺ א ݔ )ݔand there exists ܶ෨ (݁) ؔ ݈݅݉ ܶ (݁) = ݁ )ܭ(ݎݐݔܧ א ݁. ՜ஶ

෩ ()ܭ(݁݊ܥ א ݔ )ݔ, where )ܭ(݁݊ܥis the convex cone Then ݔ Ȱ generated by (ܿ{ڂ))ܭ( }. Proof. It is known that for any finite dimensional compact ܭ, its convex hull ܿ )ܭ(is compact too (the proof of this assertion is based on Carathéodory’s theorem, which leads to a way of expressing ܿ )ܭ(as image of a compact (finite dimensional) subset through a continuous mapping). Let ݉ be the dimension of the linear variety generated by ( ܭand by ܿ ))ܭ(and )ܭ(ܿ א ݔ. Assume that ݉ 2. Due to Carathéodory’s theorem, there exist at most ݉ + 1 extreme points in the compact (convex) ାଵ ߙ = 1, subset ܿ)ܭ(, say ݁ଵ , … , ݁ାଵ and {ߙଵ , … , ߙାଵ } [ ؿ0, λ), σୀଵ ାଵ such that = ݔσୀଵ ߙ ݁ . Also, it is known that any extreme point of ܿ)ܭ( is (an extreme) point of ܭ. From hypothesis we infer that ାଵ

ାଵ

ାଵ

ܶ ( = )ݔ ߙ ܶ ൫݁ ൯ ՜ ߙ ܶ෨ ൫݁ ൯ = ߙ ݁ = ݔ, ୀଵ

ୀଵ

݊ ՜ λ.

ୀଵ

Thus, there exists ܶ෨ ()ܭ(ܿ א ݔ ݔ = )ݔ. On the other side, the positive cone ܺା of the space ܺ is closed and we have assumed that Ȱ ( )ݔെ ܶ (ܺ א )ݔା for all ܺ א ݔand all ݊ אԳ. Passing to the limit, one obtains ෩ ( )ݔെ ܶ෨( = )ݔȰ ෩ ( )ݔെ ܺ א ݔା ֞ )ܭ(ܿ א ݔ Ȱ ෩ Ȱ( )ݔ )ܭ(ܿ א ݔ ݔ.

(6.6)

(Since ܶ෨ is the pointwise limit of affine operators, it is affine on ܿ ;)ܭ(a Hahn-Banach argument shows that it has an affine extension from the whole space ܧto ܧ. We denote this extension by ܶ෨ too). Recall that if 0 is not an element of ܿ)ܭ(, then [ א ߙ ;ݔߙ{ = )ܭ(݁݊ܥ ؔ ܥ0, λ), })ܭ(ܿ א ݔ

(6.7)

It is easy to see that in this case: ( ת ܥെ{ = )ܥ0 }. Now (6.6) and (6.7) yield

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139

෩ (ߙߙ = )ݔȰ ෩ ( )ݔ ߙ[ א ߙ ݔ0, λ), ֞ )ܭ(ܿ א ݔȰ ෩ ( )ݓ ܥ א ݓ ݓ. Ȱ If 0 ߲ א൫ܿ)ܭ(൯\݅ݎ൫ܿ)ܭ(൯, then ܥcould satisfy the condition ת ܥ (െ{ = )ܥ0 }, or ( ת ܥെ )ܥmight be a nonzero vector subspace (here ݅ݎ൫ܿ)ܭ(൯ is the relative interior of ܿ))ܭ(. When 0 ݅ݎ א൫ܿ)ܭ(൯, ܥis a ݉ െ dimensional vector subspace of ܺ. In both these last two cases, the conclusion of the theorem still holds true, following the same proof as in the ILUVWFDVH7KLVFRQFOXGHVWKHSURRIƑ Remark 3.1. Assume now that is not an element of ܿ)ܭ(. Then there exists a strictly positive linear continuous form ܶ on ܺ endowed with the order relation defined by ܥ, such that ԡܶԡ = 1, and a constant ߚ > 0 with inf

௫א()

ܶ( )݁(ܶ = ߚ = )ݔfor some ݁ ݎݐݔܧ א൫ܿ)ܭ(൯.

Indeed, denote by ݀ ؔ ݀൫0 , ܿ)ܭ(൯ > 0, ܸ ؔ ܤௗబ (0 ) = {א ݔ ܺ; ԡݔԡ < ݀ }. Then ܸ is a convex open neighborhood of the origin, which does not intersect ܿ)ܭ(. By geometric form of Hahn-Banach theorem (namely by a separation theorem), there exists a closed hyperplane separating ܸ and ܿ)ܭ(, and not intersecting ܸ i. e. there exists a linear continuous functional ܶ on ܺ such that 0 < )ܸ(ܶݑݏ ߚ ݂݅݊ܶ൫ܿ)ܭ(൯. In particular, ܶ( )ݔ ߚ > 0 > )ݓ(ܶ ֜ )ܭ(ܿ א ݔ0 {\ܥ א ݓ0 }. Whence ܶ is strictly positive and for any ߛ > 0, the set = ܤ { }ߛ = )ݔ(ܶ ;ܥ א ݔis a compact base for ܥ. Now scaling ܶ by a positive scalar we obtain a new strictly continuous positive form (which we also denote by ܶ) with the special property 0 < ݀ = )ܸ(ܶݑݏ = ݂݅݊ܶ൫ܿ)ܭ(൯, ܶ(݀ < )ݔ ܸ א ݔ. It results ݔ ൰ < ݀ > ߝ0, ֜ ܺ א ݔ ԡݔԡ + ߝ |ܶ( |)ݔ ԡݔԡ, ֜ ܺ א ݔԡܶԡ 1.

ܶ ൬݀

Let )ܸ߲( ת )ܭ(ܿ א څݔbe such that ܶ(݀ = ) څݔ . Then ԡ څݔԡ = ݀ and ௫ ܶ ቀԡ௫ څԡቁ = 1. Thus ԡܶԡ = 1. Consider the hyperplane ݀ = )ݔ(ܶ ;ݔ{ ؔ ܪ } څ

and the base ܥ ת ܪ = ܤ. Then the distance

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Chapter Six

݀(0 , = )ܪ

|ܶ(0) െ ݀ | = ݀ = ݀൫0 , ܿ)ܭ(൯, ԡܶԡ

as expected ( ܪand ܤare separating ݈ܿ(ܸ) and ܿ)ܭ(, but they are “tangent” to both these closed convex subsets). If ܺ is a real Hilbert space and ܶ has the properties from above, then څݔis the orthogonal (or metric) projection of 0 to ܿ)ܭ(. Consequently, ԡ څݔԡ = ݀൫0 , ܿ)ܭ(൯ = ݀ ,

څݔ٣ ܪ

Having in mind the idea of the proof of Riesz representation theorem for linear continuous forms on a Hilbert space, it results that ܶ is represented by a vector which is collinear to څݔ. Since ԡܶԡ = 1, we have to normalize څݔ. It results ܶ( ݔ, ܺ א ݔ > ݔ. ԡ څݔԡ ݀

4. Remarks on the subdifferentials of continuous sublinear operators Remark 4.1 Let ܺ be a Banach space, ܻ an order complete Banach space, ܲ: ܺ ՜ ܻ a continuous sublinear operator. Then for each ݔ ܺ אand any ܶ ߲ א௫బ ܲ we have ܶ(ݔ ) = ܲ(ݔ ), ܶ ߲ܲ א. Consequently, ߲ܲ = ߲ ܲ = ڂ௫బ אா ߲௫బ ܲ. Indeed, if ܶ ߲ א௫బ ܲ, then, by definition, ܲ( )ݔെ ܲ(ݔ ) ܶ( )ݔെ ܶ(ݔ ) for all ܧ א ݔ. Writing this for ݔݎinstead of ݔ, [ א ݎ0, λ), one obtains (via positively homogeneity): ݎ൫ܲ( )ݔെ ܶ()ݔ൯ ܲ(ݔ ) െ ܶ(ݔ ), ܺ א ݔ. If = ݎ0 we infer that ܶ(ݔ ) ܲ(ݔ ). On the other hand, dividing by > ݎ0 and passing to the limit as ݎ՜ λ we obtain ܲ( )ݔ ܶ()ݔ, ܺ א ݔ. In particular, we have obtained ܶ(ݔ ) = ܲ(ݔ ) and ܶ( )ݔ ܲ()ݔ, ܺ א ݔ. The conclusion follows. For notations used in this remark and detailed information see [39] or any othe reference concerning subdifferentails of convex operators.

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141

Remark 4.2 Let ܺ, ܻ be Banach lattices and , ܲ: ܺ ՜ ܻ a continuous sublinear operator. If ܶ: ܺ ՜ ܻ is a linear operator such that ܶ ܲ on ܺ, then ԡܶԡ ԡܲԡ in the following two cases at least: 1) ܲ is isotone; 2) ܲ is symmetric (ܲ((ܲ = )ݔെ)ܺ א ݔ )ݔ. Indeed, in case 1), if ܲ is isotone on ܺ, then ܶ( )ݔ ܲ( )ݔ ܲ(|)|ݔ, െܶ((ܶ = )ݔെ )ݔ ܲ(െ )ݔ ܲ(|)|ݔ implies |ܶ( |)ݔ ܲ(|)|ݔ, ܺ א ݔ, which further yields ԡܶ()ݔԡ ԡܲ(|)|ݔԡ ԡܲԡԡݔԡ, ฺ ܺ א ݔԡܶԡ ԡܲԡ. In case 2), assuming that ܲ is symmetric, we obtain ܶ( )ݔ ܲ()ݔ, െܶ((ܶ = )ݔെ )ݔ ܲ(െฺ )ݔ(ܲ = )ݔ |ܶ( |)ݔ ܲ( ฺ )ݔԡܶ()ݔԡ ԡܲ()ݔԡ, ฺ ܺ א ݔԡܶԡ ԡܲԡ. The preceding remark is still valid for convex symmetric operators ܲ: ܺ ՜ ܻ verifying ݑݏԡܲ()ݔԡ < ܯλ, ܲ() = .

ԡ௫ԡஸଵ

In this case, if ܶ is linear, dominated by ܲ on the entire domain space ܺ, then ԡܶԡ ܯ. Indeed, similar to the case 2) from above, we have: ԡܶ()ݔԡ ԡܲ()ݔԡ, ܺ א ݔ. This leads to ԡܶԡ = ݑݏԡܶ()ݔԡ ݑݏԡܲ()ݔԡ ܯ. ԡ௫ԡஸଵ

ԡ௫ԡஸଵ

Examples of such convex operators which are not sublinear can be constructed defining ܲ( = )ݔԡݔԡ ݕଵ , ܺ א ݔ, ( א 1, λ), where ݕଵ ܻ אା , ݕଵ ് . For the above operator, the constant ܯequals ԡݕଵ ԡ. Other examples of such convex operators that are not sublinear can be obtained forming linear combinations with positive coefficients of operators emphasized by the last formula.

CHAPTER SEVEN ON NEWTON’S METHOD FOR CONVEX FUNCTIONS AND OPERATORS AND A CONNECTION TO CONTRACTION PRINCIPLE

1. Introduction General results on a version of global Newton’s method for convex increasing or decreasing functions and operators, as well as afferent examples and applications are recalled. Connection to the contraction principle is discussed in detail and applied to approximation of ܣଵΤ , where ܣis a positive invertible symmetric operator acting on a finite dimensional Hilbert space and > 1 is a real number. Two numerical examples for 2 × 2 symmetric matrices with real coefficients are given. Some other nonlinear matrix or scalar equations are solved approximately. The strength of the method consists in its global character, while the weakness is that it is applicable only for convex functions and operators. The aim of this chapter is to emphasize a global Newton’s like method, which works only for increasing (or decreasing) convex functions and operators. The present chapter is based on the article [3] and on the review paper [56]. The theoretical results are recalled and numerous examples are illustrating the way of applying them. The connection to contraction principle plays a central role. Namely, we approximate the positive root ܷ ଵΤఈ of an arbitrary symmetric positive operator ܷ acting on the Hilbert space ܪ, with the spectrum ߪ(ܷ) ( ؿ0, ), when ߙ ( א1, ) (see [3], [56]). The approximation of the solution under attention is done by means of contraction principle, applied to a contraction operator (defined by means of Newton’s method), of contraction constant ( െ 1)Τ. If ܸ is the positive solution under attention, then clearly ܸ verifies the equation ܲ(ܷ) ؔ ܸ ఈ െ ܷ = . Some related numerical examples are given as well. Recall that Newton’s method can be used to approximate iteratively the solution of the equation ܲ( = )ݔ0,

On Newton’s Method for Convex Functions and Operators and a Connection to Contraction Principle

143

where P is a function or operator satisfying certain conditions (see the following results and examples). In some cases, the iteration ݔାଵ = ߮(ݔ ), ݊ אԳ of Newton’s method can be done by means of a contraction mapping ߮. In such cases, the evaluations of the norms of the errors given by contraction principle could be smaller than those ensured by Newton’s method. For example, if from the above example is close to 1, then the second approximation of the solution furnished by contraction principle is good enough (see the last result and the last example of this chapter). Being given a Hilbert space ܪ, and a symmetric linear operator ܷ from ܪto ܪ, the construction of a commutative algebra ܻ = ܻ(ܷ), (which is also an order complete Banach lattice) of symmetric operators acting on ܪ, is studied in detail in [17] and is reviewed below. This space of symmetric operators (in particular of symmetric matrices) plays a central role in the present work. The rest of this chapter is organized as follows. Section 2 mentions briefly the methods applied in the sequel. Section 3 is devoted to the main known results on the subject and is divided into subsections. Section 4 concludes the paper.

2. Methods The following methods are used along this chapter: 1) Newton method for convex increasing (or decreasing) functions and operators. 2) Elements of theory of symmetric operators acting on a Hilbert space. Namely, let ܪbe an arbitrary Hilbert space and ܸ a self-adjoint operator acting on ܪ. Define ܻଵ (ܸ) = {ܷ }ܷܸ = ܸܷ ;࣭ א, ܻ = ܻ(ܸ) = {ܹ ܻ אଵ ; ܹܷ = ܷܹ ܻ א ܷଵ } ܻା = {ܹ ݄ܹ < ;ܻ א, ݄ > 0 }ܪ א ݄. Then ܻ is clearly a commutative real algebra of symmetric operators. It is also an order complete real Banach lattice (for details, see [17]). Here ࣭ is the real ordered space of all self-adjoint operators acting on ܪ. 3) Contraction principle and related successive approximation method.

144

Chapter Seven

3. Main Text 3.1. General type results Let ܺ be a ߪ െ order complete vector lattice, endowed with a solid (| |ݔ | ฺ |ݕԡݔԡ ԡݕԡ) and െ continuous norm (ݔ ՜ ௗ ฺ ݔ ԡݔ െ ݔԡ ՜ 0). Let ܻ be a normed vector space, endowed with an order relation defined by a closed convex cone. For ܽ, ܾ ܺ א, ܽ < ܾ, we denote [ܽ, ܾ] = { ܽ ;ܺ א ݔ ݔ ܾ}. Let ܲ ܥ אଵ ([ܽ, ܾ], ܻ). In most of our applications, we have ܺ = ܻ, where ܻ is the order complete Banach lattice of self-adjoint operators, that is also a commutative algebra discussed in [17], p. 303-305 (see also Section 2 above). Theorem 3.1. Additionally assume that for each ܽ[ א ݔ, ܾ], ܲ[ ƍ (ି])ݔଵ א ܮା (ܻ, ܺ) and that ܽ ݔ ܾ ฺ ܲԢ(ܽ) ܲԢ( )ݔ ܲԢ(ܾ). If ܲ(ܽ) < 0 and ܲ(ܾ) > 0, then there exists a unique solution څ ݔof the equation ܲ( = )ݔ0, where = څ ݔinf ݔ = lim ݔ , ݔ = ܾ, ݔାଵ

ିଵ

= ݔ െ ൫ܲԢ(ݔ )൯ ൫ܲ(ݔ )൯, ݇ אԳ.

(7.1)

Moreover, we have ԡݔ െ څ ݔԡ ܽ < ܾ < څ ݔ, ԡ(ܲᇱ (ܽ))ିଵ ԡ ή ԡܲ(ݔ )ԡ ՜ 0, ݇ ՜ λ. Proof. Using induction upon ݇, we prove that ܲ(ݔ ) 0, ݔାଵ ݔ א ݇Գ.

(7.2) (7.3)

The last relations (7.1) and the convexity of ܲ yield: ܲ(ݔ ) = ܲ(ܾ) > 0, ܲ(ݔାଵ ) ܲ(ݔ ) + ൫ܲԢ(ݔ )൯(ݔାଵ െ ݔ ) = ܲ(ݔ ) െ ܲ(ݔ ) = . Hence ܲ(ݔ ) for all ݇ אԳ. These relations lead to ିଵ

ݔାଵ െ ݔ = െ൫ܲԢ(ݔ )൯ ൫ܲ(ݔ )൯ 0 ฺ ݔାଵ ݔ א ݇Գ. We derive the following useful relations

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ܲ(ܽ) < , െܲ(ݔ ) ฺ (ܲᇱ (ݔ ))ିଵ ൫ܲ(ܽ) െ ܲ(ݔ )൯ (ܲᇱ (ݔ ))ିଵ ൫ܲԢ(ݔ )(ܽ െ ݔ )൯ = ܽ െ ݔ ฺ ݔ ܽ א ݇Գ. Using the hypothesis on the space ܺ, there exists څ ݔdefined by the first relations (7.1), and from (7.3) we infer that the sequence (ݔ ) is decreasing. Passing to the limit in the recurrence relations (7.1) one obtains ିଵ

൫ܲԢ() څ ݔ൯ ൫ܲ() څ ݔ൯ = ฺ ܲ( = ) څ ݔ. On the other hand, from the assumptions on the positivity of ܲ(ܾ), െܲ(ܽ) and from the definition of څ ݔwe infer that ܽ < ܾ < څ ݔ, In order to prove (7.2), one uses the convexity once more: ܲ(ݔ ) = ܲ(ݔ ) െ ܲ( ) څ ݔ ܲԢ(ݔ() څ ݔ െ ) څ ݔ ܲԢ(ܽ)(ݔ െ ฺ ) څ ݔ ିଵ

൫ܲԢ(ܽ)൯ ܲ(ݔ ) ݔ െ څ ݔ ฺ ԡݔ െ څ ݔԡ ିଵ

ቛ൫ܲԢ(ܽ)൯ ቛ ή ԡܲ(ݔ )ԡ א ݇Գ. Thus (7.2) is proved. The uniqueness of the solution follows quite easily: = ܲ(ݔଵ ) څെ ܲ(ݔଶ ) څ ܲᇱ (ݔଶݔ() څଵ څെ ݔଶ ฺ ) څ0 ିଵ ൫ܲᇱ (ݔଶ) څ൯ ൫ܲԢ(ݔଶݔ() څଵ څെ ݔଶ) څ൯ = ݔଵ څെ ݔଶ څ. Similarly, we obtain ݔଶ څെ ݔଵ څ, hence ݔଵݔ = څଶ څ. The proof is complete. Ƒ The corresponding statement for convex decreasing operators holds. ିଵ

Theorem 3.2. Assume that for any ܽ[ א ݔ, ܾ] there exists ൫ܲԢ()ݔ൯ that ିଵ

െ൫ܲԢ()ݔ൯ (ܻା ) ܺ كା and ܽ ݔ ܾ ฺ ൫ܲԢ()ݔ൯

ିଵ

such ିଵ

൫ܲԢ(ܾ)൯ .

If ܲ(ܽ) > 0, ܲ(ܾ) < 0, then there exists an unique solution x ]א څa, b[ of the equation ܲ( = )ݔ0. x څbeing given by x = څsup x୩ = lim ݔ , ݔ = ܽ, ݔାଵ ୩

ିଵ

= ݔ െ ൫ܲԢ(ݔ )൯ ൫ܲ(ݔ )൯, ݇ אԳ.

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Moreover, the sequence ሺݔ ሻ is increasing and the convergence rate is given by the inequalities ିଵ

ԡݔ െ څ ݔԡ ቛ ൫ܲԢሺܾሻ൯ ቛ ή ԡܲሺݔ ሻԡ, ݇ אԳ.

3.2. Direct consequences During this Section we mention some applications of the general theorems of Section 2. The difficulties consist only in technical details concerning verifying conditions from general theorems. That is why we do not prove all the statements. Theorem 3.3. Let ܪbe a finite dimensional Hilbert space and ܺ ൌ ܻ ൌ ܻሺܸሻ the order complete Banach lattice and commutative algebra defined in [17], p. 303-305 and recalled in Section 2 above. Let ܤ ܺ אା , ݆ אሼ0,1, … , ݊ሽ, ܤ , ܤ be such that

ܤ ൏ ܤ , ୀଵ

and ݊ܤ ܷ ିଵ ڮ 2ܤଶ ܷ ܤଵ is invertible for all ܷ אሾ0, ܫሿ. Then there ഥ 0൏ܷ ഥ ൏ ܫ. such that exists a unique ܷ, ഥ ڮ ܤଵ ܷ ഥ െ ܤ ൌ , ܤ ܷ and this solution verifies in particular the relation

ഥԡ ቯ ܤ െ ܤ ቯ. ԡ ܫെ ܷ ୀଵ

Proof. The space ܺ is an order complete Banach lattice and a commutative real algebra of symmetric operators. Let ܲ: ሾ, ܫሿ ܺ ؿା ՜ ܺ, be defined by

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ܲ(ܷ) = ܤ ܷ + ڮ+ ܤଵ ܷ െ ܤ One can show that the operator ܲ , ܲ (ܷ) = ܷ , is convex on ܺା . Now it follows easily that ܲ is also convex on ܺା , since all the coefficients ܤ ܺ אା and all the operators in ܺ are permutable. On the other hand, we have ܲᇱ (ܷ)(ܸ) = (݊ܤ ܷ ିଵ + ڮ+ 2ܤଶ ܷ + ܤଵ )ܸ ฺ [ܲᇱ (ܷ)]ିଵ ൫ܸ෨ ൯ = (݊ܤ ܷ ିଵ + ڮ+ 2ܤଶ ܷ + ܤଵ )ିଵ ܸ෨ , ܸ෨ ܺ א, ܷ [ א, ]ܫ We also have [ܲᇱ (ܷ)]ିଵ 0, 0 ܷ ܲ ฺ ܫƍ (0)ܸ = ܤଵ ܸ (݊ܤ ܷ ିଵ + ڮ+ 2ܤଶ ܷ + ܤଵ )ܸ (݊ܤ + ڮ+ 2ܤଶ + ܤଵ )ܸ ฺ ܲᇱ (0) ܲᇱ (ܷ) ܲᇱ ([ א ܷ )ܫ0, ]ܫ Due to the hypothesis, we infer that

ܲ(0) = െܤ < 0, ܲ( = )ܫ ܤ െ ܤ > 0. ୀଵ

Hence all requirements of Theorem 3.1 are accomplished. It follows that there exists a unique solution ഥ ܷ ]א, [ܫof the equation ܲ(ܷ) = that verifies the following relation (for ݇ = 0 in Theorem 3.1) ԡ ܫെ ܷԡ ԡ[ܲᇱ (0)]ିଵ ԡ ή ԡܲ()ܫԡ = ԡܤଵିଵ ԡ ή ԡܤ + ڮ+ ܤଵ െ ܤ ԡ. This ends the proof. Ƒ Theorem 3.4. Let ܪ, ܺ be as in the Theorem 3.3, ߙ > 1, ܺ א ܤsuch that ഥ ]א, ܺ ؿ[ܫ the spectrum ߪ()ߙ(݈݊]ؿ )ܤ, λ[. There is a unique solution ܷ of the equation ݁ )ܷܤ(ݔെ ߙ = ܫ, and, in particular, this solution verifies the relation ഥԡ ԡି ܤଵ ԡ ή ԡexp( )ܤെ ߙܫԡ. ԡ ܫെ ܷ

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The next result is an application of the scalar version of Theorem 3.2, when ܺ = ܻ = Թ. Proposition 3.5. Let ߙ, ߚ, ߛ > 0 be such that 1 െ ߙ ή ݁(ݔെߙ) െ ߚ < ߛ < 1. Then there exists a unique solution ]א څ ݔ0,1[ of the equation ݁(ݔെߙ )ݔെ ߚ ݔെ ߛ = 0, and we have 0 < څݔ

1െߛ ՜ 0, ߙ ή ݁(ݔെߙ) + ߚ

ߛ ՛ 1.

Theorem 3.6. Let ܪbe a finite dimensional Hilbert space, and ܺ = ܻ = ܻ(ܸ) the space defined in [17], p. 303-30 and recalled above, in Section 2. Let ܣ, ܤ, ܺ א ܥbe such the spectrums of ܣand ܤare contained in ]0, f[. Assume also that ݁(ݔെ )ܣെ ܫ < ܥ < ܤ. ഥ ]א, [ܫof the equation Then there exists a unique solution ܷ ݁(ݔെ )ܷܣെ ܷܤെ = ܥ and the following estimation holds true ഥԡ ԡ( ܣή ݁(ݔെ )ܣ+ ି)ܤଵ ԡ ή ԡ ܫെ ܥԡ ՜ 0, ܥ՜ ܫ. ԡܷ Proposition 3.7. There is a unique solution ]א څ ݔ3/2,2[ of the equation 2 ݔଷ െ 4 ݔଶ + 1 = 0, and this solution verifies 27 < < څ ݔ2. 152 Theorem 3.8. Let ܸ be a symmetric operator acting on a finite dimensional Hilbert space, with the spectrum ߪ(ܸ) [ ك3Τ2 , 2]. Let ܺ = ܻ(ܸ) be the ഥ ܺ אsuch space defined in Section 2. Then there exists a unique operator ܷ that 2െ

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ഥ)ଷ െ 4(ܷ ഥ)ଶ + = ܫ, 2(ܷ and the spectrum of this operator satisfies the following relation ഥ) ]ؿ2 െ ߪ(ܷ

27 , 2[ . 152

3.3. Approximating Τࢻ , ࢻ > 1; connection to contraction principle Let ܪbe a finite dimensional Hilbert space, ܸ a symmetric operator acting on ܪ, with the spectrum ߪ(ܸ) ]ؿ0, λ[. ܺ = ܻ(ܸ) the associated commutative algebra and order complete Banach lattice defined in Section 2 and discussed in [17], p. 303-305. We denote ߱ = inf < ܸ݄, ݄ > ,

ȳ = ݄ܸ < ݑݏ, ݄ > .

ԡԡୀଵ

ԡԡୀଵ

Theorem 3.9. Let ܸ be as above, ܸ }ܫ{݊ܽܵ ב, ߙ אԹ, ߙ > 1. There exists a unique operator Τ

Τ

ܷఈ ߱]אଵ ఈ ܫ, ȳଵ ఈ [ܫ such that ܷఈఈ െ ܸ = 0, and this solution satisfies the relations: 1

Τ

ฮȳଵ ఈ ܫെ ܷఈ ฮ

ԡȳ ܫെ ܸԡ, ฮܷఈ െ ߱ଵΤఈ ܫฮ

(ఈିଵ)Τఈ ߙ߱ (ఈାଵ)Τఈ ȳ

ߙ

ԡ߱ିଵ ܫെ ܸ ିଵ ԡ .

Corollary 3.10. With the above notations and assumptions, we have ݈݃ȳ െ ݈݃ɘ

1 ԡȳ ܫെ ܸԡ + ȳ ԡ߱ିଵ ܫെ ܸ ିଵ ԡ. ɘ

Remark 3.1 If in the recurrence relation of Newton’s method ିଵ

ݔାଵ = ߮(ݔ ), ߮( ݔ = )ݔെ ൫ܲᇱ ()ݔ൯ ൫ܲ()ݔ൯,

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the mapping ߮ is a contraction, then the rate of convergence of the sequence (ݔ ) is given by contraction principle. Next, we show that this is the case of the operator ܲ(ܷ) = ܷ ఈ െ ܸ, which leads to the positive solution ܷ = ܷఈ = ܸ ଵΤఈ as a root of the equation ܲ(ܷ) = . Theorem 3.11. (See also [3]). Let ܺ, ߙ, ܸ be as above. Then the Newton recurrence for the equation ܲ(ܷ) = ܷ ఈ െ ܸ = is Τ

ܷ = ߗଵ ఈ ܫ, ܷାଵ =

ఈିଵ ఈ

ଵ

ܷ + ܷିఈାଵ ܸ, ݇ אԳ. ఈ

The convergence rate for ܷ ՜ ܸ ଵΤఈ is given by ฮܷ െ ܸ ଵΤఈ ฮ ൬

ߙ െ 1 ଵΤఈ ൰ ȳ ԡ ܫെ ȳିଵ ܸԡ, ݇ אԳ. ߙ

Proof. Newton’s sequence for the convex operator ܲ is Τ

ܷ = ܾ = ߗଵ ఈ ܫ, ܷାଵ = ܷ െ (ߙܷఈିଵ )ିଵ (ܷఈ െ ܸ) = ܷ െ

1 ିఈାଵାఈ 1 ିఈାଵ ߙെ1 1 ܷ + ܷ ܸ= ܷ + ܷିఈାଵ ܸ ߙ ߙ ߙ ߙ = ߮(ܷ ), ݇ אԳ.

Let = ܯ൛ܷ ܷ ;ܺ א ܸ ଵΤఈ ൟ. Here the root is obtained by the aid of functional calculus for ܸ. Clearly, ܯ is closed in ܺ, hence it is complete. Let ߮: ܯ՜ ܺ be defined by ߮(ܷ) =

ߙെ1 1 ܷ + ܷ ିఈାଵ ܸ, ߙ ߙ

ܷ ܯ א.

A straightforward computation shows that ߮൫ܸ ଵΤఈ ൯ = ܸ ଵΤఈ . First we show that ߪ(ܷ) ( ؿ0, λ) ฺ ߮(ܷ) ;ܯ א

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(in particular this proves that ߮()ܯ ك )ܯ. One can show that ߮ is convex on the subset of all operators in ܺ having the spectrum contained in the positive semiaxis. In particular, ߮ is convex on ܯ. Direct computations yield ߮(ܷ) ߮൫ܸ ଵΤఈ ൯ + ߮ ᇱ ൫ܸ ଵΤఈ ൯൫ܷ െ ܸ ଵΤఈ ൯ = ܸ ଵΤఈ . Thus ߮(ܷ) for all ܷ with spectrum ߪ(ܷ) ( ؿ0, λ). Now we prove that ఈିଵ . Precisely we ߮: ܯ՜ ܯis a contraction, with contraction constant = ݍ ఈ prove that ԡ߮Ԣ(ܷ)ԡ

ఈିଵ ఈ

ܯ א ܷ.

Indeed, we have ฮ߮ ᇱ() ฮ =

ߙെ1 ԡ ܫെ ܷ ିఈ ܸԡ; ܷ ܷ ฺ ܯ אఈ ܸ ฺ ߙ ܫെ ܷ ିఈ ܸ ฺ

ԡ ܫെ ܷ ିఈ ܸԡ ԡܫԡ = 1 ฺ ԡ߮Ԣ(ܷ)ԡ

ߙെ1 , ߙ

ܷ ܯ א.

Now the conclusion on ߮ being a contraction follows by a standard differential calculus argument. Application of contraction theorem and an elementary computation shows that ฮܷ െ ܸ ଵΤఈ ฮ =൬

ݍ ԡܷ െ ߮(ܷ )ԡ 1െ ݍ

ߙ െ 1 ଵΤఈ ൰ ȳ ԡ ܫെ ȳିଵ ܸԡ, ݇ אԳ. ߙ

7KHSURRILVFRPSOHWHƑ Numerical examples 2 2 ଵΤ() , evaluating the norm of the error. ቁ 2 5 Notice that ݈݃6 is not an integer.

1) We approximate ቀ

Therefore, simple recurring approximating sequences of the matrix from above might be difficult to find. We apply the previous Theorem 3.11.

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2 2 ቁ, 2 5 ଶ applying Թ onto itself. The spectrum of ܸ is ߪ(ܸ) = {1,6} ( ؿ0, ), and ܸ is not contained in ܵ)ܫ(݊ܽ. Hence all conditions of Theorem 3.11 are accomplished. Applying the latter theorem, one obtains the following two approximations of ܸ ଵΤ : Consider the linear symmetric operator ܸ defined by the matrix ቀ

ଵΤ

ܷ = ȍ ܷଵ =

= ܫ6ଵΤ ܫ݁ = ܫ,

݈݃6 െ 1 1 ଵି ܷ + ܷ ܸ ݈݃6 ݈݃6 ݁ 6݈݃6 െ 4 ൬ = 2 6݈݃6

2 ൰. 6݈݃6 െ 1

For the evaluation of the norm of the error corresponding to the second approximation ܷଵ , one applies the last relation in the statement of Theorem 3.11, where ݇ = 1, ߙ = ݈݃6. One deduces: ฮܷଵ െ ܸ ଵΤ ฮ ൬1 െ

݈݃6 െ 1 ଵΤ ԡ ܫെ (1Τ6)ܸԡ = 6 ݈݃6

1 1 2Τ3 ൰ ݁ԡܤԡ, ܫ = ܤെ ܸ = ൬ െ 1Τ3 ݈݃6 6

െ 1Τ3 ൰. 1Τ6

The spectrum of the matrix ܤis ߪ({ = )ܤ0, 5Τ6} [ ؿ0, ) and ܤis symmetric, so that ԡܤԡ = ȍ = 5Τ6. Thus ฮܷଵ െ ܸ ଵΤ ฮ ቀ1 െ

ଵ

ହ

ଵ

ଵ.଼

ቁ ݁ ቀ1 െ

ହ

ଵ

ଵଷହ

ቁ ή 2.72 ή = 1 +

..

The conclusion is ܸ ଵΤ = ቀ

2 2

݁ 6݈݃6 െ 4 2 ଵΤ() ቁ ൎ ൬ 2 5 6 ή ݈݃6 = ܷଵ .

2 ൰ 6݈݃6 െ 1

and the norm of the error in the latter approximation is smaller than 1 + భΤ

ଵ ଵଷହ

.

2 2 ଵΤ൫ ൯ ቁ , ݊ אԳ, ݊ 2, evaluating the norm of 2 5 the error. Using the notations and some of the results for preceding Example

2) One approximates ቀ

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1), and applying Theorem 3.11 to the ܺ = ܻ = ܻ(ܸ) defined in Section 2, associated to the operator (or matrix) ܸ, we find: ܷ (݊) = 6ଵΤ൫

1

݊

൭ ଵΤ

భΤ ൯

ܫ, ܷଵ (݊) ଵିభΤ ݊ଵΤ െ 1 ଵΤ൫భΤ൯ 1 ଵΤ൫భΤ൯ = 6 ܫ + ܸ ή ቀ6 ܫቁ Τ Τ ݊ଵ ݊ଵ =

൫݊ଵΤ െ 1൯6ଵΤ൫

భΤ ൯

2 ή 6(ଵି

+ 2 ή 6(ଵି

భΤ )ΤభΤ

భΤ )ΤభΤ

2 ή 6(ଵି ൫݊ଵΤ െ 1൯6ଵΤ൫

భΤ ൯

భΤ )ΤభΤ

+ 5 ή 6(ଵି

భΤ )ΤభΤ

భΤ

2 2 ଵΤ൫ ൯ ቁ Hence ቀ ൎ ܷଵ (݊) and the norm of the error follows from the 2 5 last evaluation in the statement of Theorem 3.11: ቛܸ ଵΤ൫

భΤ ൯

െ ܷଵ (݊)ቛ

݊ଵΤ െ 1 ଵΤ൫భΤ൯ 1 6 ฯ ܫെ ܸฯ. Τ ଵ 6 ݊ 1

Applying the result from Example 1) for ቛ ܫെ ܣቛ, the latter relation 6 further yields ቛܸ ଵΤ൫

భΤ ൯

݊ଵΤ െ 1 ݊ଵΤ 6 5൫݊ଵΤ െ 1൯ ՜ 0, ݊ ՜ λ.

െ ܷଵ (݊)ቛ

5

ଵିଵΤ൫భΤ൯

ή

The conclusion is that the second approximation ܷଵ (݊) is good enough, for large ݊.

൱.

CHAPTER EIGHT ON A CLASS OF SPECIAL FUNCTIONAL EQUATIONS

We start by recalling some of our earlier results on the construction of a nontrivial function ݂݂ defined implicitly by the equation (8.1), without using the implicit function theorem. This is the first aim of the paper. Here the function ݃݃ is given, satisfying some conditions. All these considerations work in the real case, for functions and a class of operators (for the case of operatorial equations, see the references of [57]). The second aim is to consider the complex case, proving the analyticity of the function ݂݂ defined implicitly, under the hypothesis that ݃݃ is analytic and verifies natural conditions, related to the real case. Some consequences are deduced. Finally, one illustrates the preceding results by an application to a concrete functional equation. Related examples are given, some of them pointing out elementary functions ݃݃ for which equation (8.1) leads to nontrivial solutions ݂݂ that can be expressed by means of elementary functions.

1. Introduction The equation ݃ =݂݃ל

(8.1)

where g is given, while f is the unknown function, always has the trivial solution ݂(ݔ = )ݔ,

ܦ א ݔ,

where ܦis the domain of definition for ݂. When (the nonlinear) function ݃ is firstly decreasing and then increasing (hence െ݃ firstly increases and then decreases), there exists exactly one decreasing nontrivial solution ݂ with the properties stated in Theorem 3.1 below. These equations were studied in the review paper [57] (see the references therein for the related

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155

original articles). For concrete functions ݃ one obtains special properties of the corresponding solutions ݂. The present approach allows the construction of the solutions of such functional equations, without using the implicit function theorem. The rest of the chapter is organized as follows. Section 2 emphasizes the methods applied along this work. In the first part of Section 3, we recall some known results on the subject, especially related to the real case. Then we consider the case of complex analytic functions. The general idea is that analyticity of ݃ implies the same property for ݂. In the end of Section 3, concrete functional equations are solved and simple examples when the solution can be written explicitly in terms of elementary functions are given. Section 4 concludes the chapter.

2. Methods The main methods used along this chapter are: 1) Constructing a real function ݂ defined implicitly by a given real continuous function ݃ of one real variable, under appropriate assumptions on ݃ (see Theorem 3.1, assertion (x)), without using the implicit function theorem. Deducing basic properties of the “unknown” function ݂. 2) In the case when ݃݃ is a complex analytic function of one complex variable, deducing the analyticity of ݂. Here elements of elementary complex analysis [68] are applied.

3. Results and Discussion I) The real case ഥ , ݒ < ݑ, ߙ ݑ( א, )ݒand let ݃: (ݑ, )ݒ՜ Թ be a Theorem 3.1. Let ݑ, א ݒԹ continuous function. Assume that ഥ, (a) lim ݃( = )ݔlim ݃( א ݓ = )ݔԹ ௫՝௨

௫՛௩

(b) g is strictly decreasing on (ݑ, ߙ) and strictly increasing on [ߙ, )ݒ. Then there exists ݂: (ݑ, )ݒ՜ (ݑ, )ݒsuch that

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݃(݃ = )ݔ൫݂()ݔ൯ for all ݑ( א ݔ, )ݒand f satisfies the following conditions: (i) ݂ is strictly decreasing on (ݑ, )ݒand we have lim ݂(ݒ = )ݔ, lim ݂(;ݑ = )ݔ ௫՝௨

௫՛௩

ߙ is the unique fixed point of ݂; we have ݂ ିଵ = ݂ on (ݑ, ;)ݒ f is continuous on (ݑ, ;)ݒ if ݃ ܥ א ((ݑ, )}ߙ{\)ݒ, ݊ אԳ }{ , ݊ 1, then ݂ א ܥ ((ݑ, ;)}ߙ{\)ݒ (vi) if g is differentiable on (ݑ, }ߙ{\)ݒ, so is f. ഥ, (vii) If ݃ ܥ אଶ (ݑ, )ݒ, ݃ᇱᇱ (ߙ) ് 0 and there exists ߩଵ = lim ݂Ԣ( א )ݔԹ

(ii) (iii) (iv) (v)

௫՜ఈ

then ݂ ܥ אଵ ((ݑ, ܥ ת ))ݒଶ ((ݑ, )}ߙ{\)ݒand ݂Ԣ(ߙ) = െ1; ഥ (viii) if ݃ ܥ אଷ (ݑ, )ݒ, ݃ᇱᇱ (ߙ) ് 0 and there exists ߩଵ = lim ݂Ԣ( א )ݔԹ ௫՜ఈ

and ߩଶ = lim ݂ԢԢ( א )ݔԹ, then ௫՜ఈ

݂ ܥ אଶ ൫(ݑ, )ݒ൯ ܥ תଷ ൫(ݑ, }ߙ{\)ݒ൯, and ଶ

݂ ᇱᇱ (ߙ) = ߩଶ = െ ή ݃ԢԢԢ(ߙ)Τ݃ԢԢ(ߙ) ; ଷ

(ix) if ݃ is analytic at ߙ, then ݂ is derivable at ߙ and ݂ ᇱ (ߙ) = െ1; (x) let ݃ = ݃|(௨,ఈ) , ݃ = ݃|(ఈ,௩); then the following constructive formulae for ݂(ݔ ) hold true: ݂(ݔ ) = (݃ିଵ ݃ ל )(ݔ ) = ߙ[ א ݔ{ݑݏ, ݃ ;)ݒ ( )ݔ ݃ (ݔ )} ݔ ݑ( א, ߙ], and ݂(ݔ ) = (݃ିଵ ݃ ל )(ݔ ) = ݂݅݊{ݑ( א ݔ, ߙ]; ݃ ( )ݔ ݃ (ݔ )} ݔ ߙ[ א, )ݒ.

On a Class of Special Functional Equations

157

We recall the geometric meaning of the construction of ݂. If ݑ( א ݔ, }ߙ{\)ݒ, consider the horizontal passing through the point ൫ݔ, ݃()ݔ൯. Thanks to the qualities of ݃, this straight line intersects once again the graph of ݃ at exactly one point ൫ݔଵ , ݃(ݔଵ )൯ = ൫ݔଵ , ݃()ݔ൯, ݔଵ ് ݔ. We define ݂(ݔ = )ݔଵ . Then we have ݃(ݔ(݃ = )ݔଵ ) = ݃൫݂()ݔ൯, ݑ( א ݔ, }ߙ{\)ݒ, ݂(ߙ) ؔ ߙ. When ݔruns over the interval (ݑ, ߙ], ݂( )ݔruns over the interval [ߙ, )ݒ, in the decreasing sense, from ݒto ߙ When ݔruns over the interval [ߙ, ( )ݒin the increasing sense), ݂( )ݔruns over the interval (ݑ, ߙ] in the decreasing sense, from ߙ to ݑ. Let ( א0, λ) and denote by ܩthe set of all continuous functions ݃: (0, λ) ՜ (0, λ), ݃(ߙ) = 0, which are decreasing on (0, ߙ] and increasing on [ߙ, λ), such that lim ݃ ( = )ݔlim ݃( = )ݔλ. ௫՝

௫՛ஶ

For ݃, ݄ ܩ א, an interesting problem is the following one: find necessary and sufficient conditions on ݃, ݄ for the following equality holds: ݂ = ݂ , where ݂ , ݂ are the corresponding functions attached to ݃ respectively to ݄ by Theorem 3.1. The following statement is giving the answer. Theorem 3.2. Let ݃, ݄ ܩ א, ߣ ( א0, λ). Then ݃ + ݄, ߣ݃, ݃ ή ݄ are also elements of ܩand the following statements are equivalent (a) ݂ = ݂ ; (b) ݄ ݃ לିଵ = ݄ ݃ לିଵ ; (c) there exists ߰: [0, λ) ՜ [0, λ) such that ߰(0) = 0, ߰ is continuous and increasing, verifying the relation

158

Chapter Eight

݄ = ߰ ݃ ל.

II) On the analyticity of the solution. The complex case Application of the complex form of the implicit function theorem for a holomorphic function ݃ that is the extension of the real function ݃ ݃ of Theorem 3.1, might be difficult around ߙߙ. Namely considering the equation ݖ(ܨ, ݃ = )ݓ( )ݖെ ݃( = )ݓ0, we have డி డ௪

(ߙ, ߙ) = 0, lim

ᇲ (௭)

௭՜ఈ ᇲ (௪(௭))

= = lim ݓᇱ ()ݖ.

௭՜ఈ

It is worth noticing that for points from (ݑ, }ߙ{\)ݒwe can apply the implicit function theorem (over the complex field), which leads to the analyticity of ݂ at such points. Therefore, we only have to study the analyticity at ߙߙ. Theorem 3.3. Let ݃ be the extension of the function ݃ of Theorem 3.1, such that ݃ is holomorphic in a complex neighborhood of (ݑݑ, [)ݒ. Then there is a unique holomorphic solution ݂ሚ of the equation ݃ = ݃ ݂ לሚ such that ݂ሚ is the extension of ݂ of Theorem 3.1 to a complex neighborhood of (ݑ, )ݒ. Proof. From the preceding remarks, we have to prove the analyticity of ݂ሚ only at ߙ. To this end, let us write the following expansions: ݃( )ݖെ ݃(ߙ) = ( ݖെ ߙ)ଶ ቈ

݃(ଶ) (ߙ) ݃(ଶାଵ) (ߙ) ( ݖെ ߙ) + 0( ݖെ ߙ), + (2݇)! (2݇ +)!

݃൫݂()ݖ൯ െ ݃(ߙ) = (݂( )ݖെ ߙ)ଶ ቈ

݃(ଶ) (ߙ) ݃(ଶାଵ) (ߙ) (݂( )ݖെ ߙ) + 0(݂( )ݖെ ߙ) + (2݇)! (2݇ +)!

Here ݉ = 2݇ 2 is the smallest natural number, which for the derivative of order ݉ of ݃ at ߙ is not vanishing. By Taylor formula, it must be an even number, since ߙߙ is a minimum point for ݃. On the other hand, since ݃ = ݃ ݂ לሚ, elementary computations and the preceding expansions lead to

On a Class of Special Functional Equations

159

lim ݂ሚ( ߙ = )ݖቀ= ݂ሚ(ߙ)ቁ.

௭՜ఈ

Further computations yield ݃(ଶ) (ߙ) + ߮( ݖ()ݖെ ߙ) ሚ ሚ ݂ ( )ݖെ ݂ (ߙ) (2݇)! ۊ = ( ۇଶ) ݖെߙ ݃ (ߙ) ሚ ሚ + ߮ ቀ݂()ݖቁ ൫݂( )ݖെ ߙ൯ ی ( ۉ2݇)!

ଵΤଶ

,

෩ (ߙ) is a 2݇-order root where ߮ is holomorphic around ߙߙ. It follows that ݂ƍ ሚ of the unity. Using the fact that ݂ applies intervals of the real line into the ෩ (ߙ) = ݂ ƍ (ߙ) = െ1. This real line and that it is decreasing we deduce that ݂ƍ FRQFOXGHVWKHSURRIƑ In the sequel, for a complex neighborhood ܦof ߙ, we denote ܦା = {ߙ < ݖܴ݁ ;ܦ א ݖ, > ݖ݉ܫ0}, ܦି = {ߙ < ݖܴ݁ ;ܦ א ݖ, < ݖ݉ܫ0}, ܦା = {ߙ > ݖܴ݁ ;ܦ א ݖ, > ݖ݉ܫ0}, ܦି = {ߙ > ݖܴ݁ ;ܦ א ݖ, < ݖ݉ܫ0}. Corollary 3.4. There is a neighborhood ܦof ߙ such that ݂ሚ: ܦ՜ ݂ሚ()ܦ is a one- to-one mapping, we have ݂ሚ ݂ לሚ = ݅݀, and ି

ା

ି

ା

݂ሚ(ܦା ) = ቀ݂ሚ()ܦቁ , ݂ሚ(ܦି ) = ቀ݂ሚ()ܦቁ , ݂ሚ(ܦା ) = ቀ݂ሚ()ܦቁ , ݂ሚ(ܦି ) = ቀ݂ሚ()ܦቁ . Proof. The first two assertions follow by the local inversion theorem and respectively from the analytic continuation principle (݂ሚ is a holomorphic extension of ݂). The last four relations are consequences of the fact that ݂ሚ is conformal at ݔ0 = ߙ, also using the qualities of ݂ (see the comments IROORZLQJ7KHRUHP 7KHFRQFOXVLRQVIROORZƑ Corollary 3.5. There is a complex neighborhood ܹof the interval (ݑ, )ݒ such that ݂ሚ is holomorphic on that neighborhood.ܹ.

160

Chapter Eight ଵ

Corollary 3.6. The function ௭ିఈ + ൫݂ሚ( )ݖെ ߙ൯ is univalent in the open disk | ݖെ ߙ| < 1 if and only if ݂( = )ݖെ ݖ+ 2ߙ. Proof. For the only if part, assume that 1 ݂ ƍƍ (ߙ) 1 ( ݖെ ߙ)ଶ + ڮ + (݂( )ݖെ ߙ) = െ ( ݖെ ߙ) + 2 ݖെߙ ݖെߙ is univalent in the open disc | ݖെ ߙ| < 1. Then by the area theorem 14.13 [68], we should have ஶ

݊|ܽ |ଶ 1, ୀଵ

where ܽ ܽ ܽ , ݊݊݊ 1 are the coefficients of the holomorphic part of the preceding expansion. Since ܽ1 = െ1, it follows that all the others ܽ , ݊ 2, are vanishing, so that ݂ሚ( )ݖെ ߙ = െ ݖ+ ߙ, ݂ሚ( = )ݖെ ݖ+ 2ߙ. Conversely, if this last relation is verified, then a straightforward computation shows that the function 1 + (݂( )ݖെ ߙ) ݖെߙ is univalent in the unit open disk centered at ߙ. 7KLVFRQFOXGHVWKHSURRIƑ Remark 3.1. The function ݂ሚ( = )ݖെ ݖ+ 2ߙ is an extreme point of the convex set of all holomorphic functions with real coefficients = )ݖ(ݓ ଶ ஶ σஶ ୀଵ ܽ ( ݖെ ߙ) , ܽ אԹ, σୀଵ ݊ܽ 1. It is also an extreme point of the convex subset of all functions ஶ

= )ݖ(ݓ ܽ ( ݖെ ߙ) , ୀଵ ஶ

ܽ אԹ, ܽଵଶ + ܾ |ܽ | 1, ܾ 0, ݉ > 1, ݊ 2. ୀଶ

Theorem 3.7. In a small neighborhood of ߙ, we have

On a Class of Special Functional Equations

161

ᇲ

(௭) ݂ሚ( )ݖൎ ݖെ 2 ᇲᇲ (௭) .

Proof. The following relations hold true 0 = ݃൫݂()ݖ൯ െ ݃()ݖ = ݃ᇱ ( )ݖ(݂()ݖെ )ݖ+ (݃ᇱᇱ ()ݖΤ2)(݂( )ݖെ )ݖଶ + 0((݂( )ݖെ )ݖଶ ). Dividing by ݂( )ݖെ ് ݖ0 and neglecting the remained terms in ݂( )ݖെ ݖ, the conclusion follows. Ƒ Corollary 3.8. For ݃ as above and any linear bounded operator ܷ acting on a Hilbert space ܪ, with spectrum ߪ(ܷ) in a small neighborhood of ߙ ߙߙ, there is a holomorphic function ݂ሚ such that ݃ ቀ݂ሚ(ܷ)ቁ = ݃(ܷ), ݂ሚ ቀ݂ሚ(ܷ)ቁ = ܷ, ߪ(ܷ) ݖ݉ܫ{ ؿ 0} ߪ ቀ݂ሚ(ܷ)ቁ ݖ݉ܫ{ ؿ 0}, ିଵ ݂ሚ(ܷ) ൎ ܷ െ 2݃ᇱ (ܷ)൫݃ᇱᇱ (ܷ)൯ ,

for all such operators ܷ. Proof. Part of the relations follows by analytic functional calculus. For the third relation, one applies Corollary 3.4. For the last relation one uses Theorem 3.7Ƒ

III) Examples We consider the functional equation ఈ

ݔఈ ݁(ݔെߚ = )ݔ൫݂()ݔ൯ ݁ݔ൫െߚ݂()ݔ൯, > ݔ0, ߙ > 0, ߚ > 0. This equation is equivalent to the following one (ݔ݁ݔെܾݔ݁)ݔ(݂ = )ݔ൫െܾ݂()ݔ൯, > ݔ0, ܾ ؔ ߚ Τܽ > 0

(8.2)

Theorem 3.9. There exists a unique decreasing solution ݂: (0, ) ՜ (0, λ) of the equation (8.2) and this solution has the following properties:

162

Chapter Eight

(i) ݂(0 +) = λെ, ݂(λ െ) = 0+; (ii) ߙ = 1Τܾ is the unique fixed point of ݂; (݅݅݅) ݂ ିଵ = ݂ on (0, λ); (iv) the following constructive formulae for ݂( )ݔhold: ݂(ݔ ) = [ א ݔ{ݑݏ1Τܾ, λ[; ݔή ݁(ݔെܾ )ݔ ݔ ή ݁(ݔെܾݔ )}, ݔ ]א0, 1Τܾ], ݂(ݔ ) = ݂݅݊{]א ݔ0, 1Τܾ]; ݔή ݁(ݔെܾ )ݔ ݔ ή ݁(ݔെܾݔ )}, ݔ [ א1Τܾ, λ[; (v) ݂ is the restriction of a holomorphic function ݂ሚ on a complex neighborhood of (0, λ) such that ݂ሚ ᇱ (1Τܾ) = െ1, ݂ሚ ݂ לሚ = ݅݀, and ݂ሚ has the properties mentioned in Corollary 3.4. (vi) In a small neighborhood of 1Τܾ we have: ଶ ଵି௭ ݂ሚ( )ݖൎ ݖ+ ή .

ଶି௭

Proof. The function ݃( = )ݔെ ݔή ݁(ݔെܾ )ݔdecreases from 0 to ݃(1Τܾ) = െ݁ ିଵ ܾ ିଵ in the interval [0, ܾ ିଵ ] and increases from ݃(1Τܾ ) to 0 in the interval (ܾ ିଵ , λ). Hence the conclusions (i)-(iv) follow from Theorem 3.1. The function ݃ is the restriction of a holomorphic function ݃ with ݃ƍ (ܾ ିଵ ) = 0, ݃() (( = )ݖെ1) ܾ ିଵ ή ݁(ݔെܾ ݊()ݖെ ܾ)ݖ, ݊ אԳ, ݊ 1, א ݖԧ. In particular, ݃ᇱᇱ (ܾ ିଵ ) ് 0, so that one applies Corollary 3.4 that leads to the conclusion (v) of the present statement. The assertion (vi) follows from Theorem 3.77KHSURRILVFRPSOHWHƑ Example 3.1. The nontrivial solution of the equation ݁ ି௫ (1 െ ݁ ି௫ ) = ݁ ି(௫) ൫1 െ ݁ ି(௫) ൯, > ݔ0, is given by ݂( = )ݔെ݈(݃1 െ ݁ ି௫ ), > ݔ0. Its holomorphic extension is

On a Class of Special Functional Equations

163

݂ሚ( = )ݖെ݈(݃1 െ ݁ ି௭ ), ܴ݁ > ݖ0. Proof. One can prove easily that the function ݃( = )ݔെ݁ ି௫ (1 െ ݁ ି௫ ), ( א ݔ0, λ) satisfies the conditions of Theorem 3.1, where ߙ = ݈(݃2). To find the analytic expression of ݂݂, we rewrite our functional equation as ݁ ି(௫) െ ݁ ି௫ = ݁ ିଶ(௫) െ ݁ ିଶ௫ = ൫݁ ି(௫) െ ݁ ି௫ ൯൫݁ ି(௫) + ݁ ି௫ ൯ Dividing by ݁ ି(௫) െ ݁ ି௫ , (݈݃ ് ݔ2), and doing some straightforward computations, one obtains the result. This FRQFOXGHVWKHSURRIƑ Example 3.2. The unique nontrivial solution ݂ሚ of the equation ݃(݃ = )ݖ ቀ݂ሚ()ݖቁ , | < |ݖ1, ݃( = )ݖቀ

௭ ଶି௭

ଶ

ቁ ,

௭

is given by ݂ሚ( = )ݖ௭ି1. Example 3.3. The unique nontrivial solution of the functional equation ݃(݃ = )ݔ൫݂()ݔ൯, ݃( ݔ = )ݔ+

1 , ( א ݔ0, λ), ݔ

ଵ

is ݂( = )ݔ௫. A similar remark holds true for ( א ݔെλ, 0). It seems that ଵ

ଵ

this is example remains valid in complex, for ݃( ݖ = )ݖ+ ௭ , ݂( = )ݖ௭ , א ݖԧ\{0}.

Conclusions We have proved that under certain assumptions on the given function ݃, there exists a unique nontrivial solution ݂ of the functional equation (1). In case of real functions ݃ of one real variable, verifying the condition of Theorem 3.1, the nontrivial strictly decreasing solution ݂ has as unique fixed point the unique minimum point of ݃. The real case for scalar-valued

164

Chapter Eight

functions leads to generalization to functions of self-adjoint operators, by means of functional calculus (when inequalities are preserved). Along the proofs of these first results, the implicit function theorem is not used, even in the case when ݃ is smooth. In the case of real valued functions ݃, a geometric meaning of the construction of the nontrivial solution ݂ is pointed out in the comment following Theorem 3.1. This idea is basic for the construction of ݂, not only for real valued functions, but also in the context of abstract (and concrete) order complete vector lattices (see [57] and the references therein). In the case of complex-valued functions ݃, if ݃ is analytic, so is ݂. Finally, we emphasize three elementary functions ݃ for which the solution ݂ is expressible by means of elementary functions as well.

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